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LLLLLEEEEEAAAAARRRRRNNNNNIIIIINNNNNGGGGG OOOOOBBBBBJJJJJEEEEECCCCCTTTTTIIIIIVVVVVEEEEESSSSS
Often students will come across a sequence of numbers which are having a common difference,
i.e., difference between the two consecutive pairs are the same. Also another very common
sequence of numbers which are having common ratio, i.e., ratio of two consecutive pairs are the
same. Could you guess what these special type of sequences are termed in mathematics?
Read this chapter to understand that these two special type of sequences are called Arithmetic
Progression and Geometric Progression respectively. Further learn how to find out an element
of these special sequences and how to find sum of these sequences.
These sequences will be useful for understanding various formulae of accounting and finance.
The topics of sequence, series, A.P., G.P. find useful applications in commercial problems among
others; viz., to find interest earned on compound interest, depreciations after certain amount of
time and total sum on recurring deposits, etc.
66666.....11111 SSSSSEEEEEQQQQQUUUUUEEEEENNNNNCCCCCEEEEE
Let us consider the following collection of numbers-
(1) 28 , 2, 25, 27, ————————
(2) 2 , 7, 11, 19, 31, 51, —————
(3) 1, 2, 3, 4, 5, 6, ———————
(4) 20, 18, 16, 14, 12, 10, —————
In (1) the nos. are not arranged in a particular order. In (2) the nos. are in ascending order but
they do not obey any rule or law. It is, therefore, not possible to indicate the number next to 51.
In (3) we find that by adding 1 to any number, we get the next one. Here the no. next to 6 is
(6 + 1 = ) 7.
In (4) if we subtract 2 from any no. we get the nos. that follows. Here the no. next to 10 is
(10 –2 =) 8.
Under these circumstances, we say, the nos. in the collections (1) and (2) do not form sequences
whereas the nos. in the collections (3) & (4) form sequences.
Thus a sequence may be defined as follows:—
AAAAAnnnnn ooooorrrrrdddddeeeeerrrrreeeeeddddd cccccooooolllllllllleeeeeccccctttttiiiiiooooonnnnn ooooofffff nnnnnuuuuummmmmbbbbbeeeeerrrrrsssss aaaaa,,,,, aaaaa,,,,, aaaaa,,,,, aaaaa,,,,, .....................................................................................,,,,, aaaaa ,,,,, ..................................................................................... iiiiisssss aaaaa ssssseeeeeqqqqquuuuueeeeennnnnccccceeeee iiiiifffff aaaaaccccccccccooooorrrrrdddddiiiiinnnnnggggg
11111 22222 33333 44444 nnnnn
tttttooooo sssssooooommmmmeeeee dddddeeeeefffffiiiiinnnnniiiiittttteeeee rrrrruuuuullllleeeee ooooorrrrr lllllaaaaawwwww,,,,, ttttthhhhheeeeerrrrreeeee iiiiisssss aaaaa dddddeeeeefffffiiiiinnnnniiiiittttteeeee vvvvvaaaaallllluuuuueeeee ooooofffff aaaaa cccccaaaaalllllllllleeeeeddddd ttttthhhhheeeee ttttteeeeerrrrrmmmmm ooooorrrrr eeeeellllleeeeemmmmmeeeeennnnnttttt ooooofffff ttttthhhhheeeee
nnnnn ,,,,,
ssssseeeeeqqqqquuuuueeeeennnnnccccceeeee,,,,, cccccooooorrrrrrrrrreeeeessssspppppooooonnnnndddddiiiiinnnnnggggg tttttooooo aaaaannnnnyyyyy vvvvvaaaaallllluuuuueeeee ooooofffff ttttthhhhheeeee nnnnnaaaaatttttuuuuurrrrraaaaalllll nnnnnooooo..... nnnnn.....
Clearly, a is the 1st term of the sequence , a is the 2nd term, ................., a is the nth term.
1 2 n
In the nth term a , by putting n = 1, 2 ,3 ,......... successively , we get a , a , a , a , .........
n 1 2 3 4
Thus it is clear that the nth term of a sequence is a function of the positive integer n. The nth term
is also called the general term of the sequence. To specify a sequence, nth term must be known,
otherwise it may lead to confusion. A sequence may be finite or infinite.
If the number of elements in a sequence is finite, the sequence is called finite sequence; while if the
number of elements is unending, the sequence is infinite.....
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A finite sequence a , a , a , a , ................., a is denoted by {a }n and an infinite sequence a , a ,
1 2 3 4 n i i=1 1 2
a , a , ................., a ................. is denoted by {a }∞ or simply by
3 4 n , n n=1
{ a } where a is the nth element of the sequence.
n n
EEEEExxxxxaaaaammmmmpppppllllleeeee :::::
1) The sequence { 1/n } is 1, 1/ , 1/ , 1/ ,……
2 3 4
2) The sequence { ( – 1 ) n n } is –1, 2, –3, 4, –5,…..
3) The sequence { n } is 1, 2, 3,…
4) The sequence { n / (n + 1) } is 1/, 2/, 3/ , 4/, …….
2 3 4 5
5) A sequence of even positive integers is 2, 4, 6, .....................................
6) A sequence of odd positive integers is 1, 3, 5, 7, .....................................
All the above are infinite sequences.
EEEEExxxxxaaaaammmmmpppppllllleeeee:::::
1) A sequence of even positive integers within 12 i.e., is 2, 4, 6, 10.
2) A sequence of odd positive integers within 11 i.e., is 1, 3, 5, 7, 9. etc.
All the above are finite sequences.
66666.....22222 SSSSSEEEEERRRRRIIIIIEEEEESSSSS
An expression of the form a + a + a + ….. + a + ............................ which is the sum of the
1 2 3 n
elements of the sequenece { a } is called a series. If the series contains a finite number of elements,
n
it is called a finite series, otherwise called an infinite series.
If S = u + u + u + u + ……. + u , then S is called the sum to n terms (or the sum of the first
n 1 2 3 4 n n
n terms ) of the series and is denoted by the Greek letter sigma ∑.
n
Thus, S = ∑ u or simply by ∑u
n r=1 r n.
IIIIIlllllllllluuuuussssstttttrrrrraaaaatttttiiiiiooooonnnnnsssss :
(i) 1 + 3 + 5 + 7 + ............................ is a series in which 1st term = 1, 2nd term = 3 , and so on.
(ii) 2 – 4 + 8 –16 + ............................ is also a series in which 1st term = 2, 2nd term = –4 , and so on.
66666.....33333 AAAAARRRRRIIIIITTTTTHHHHHMMMMMEEEEETTTTTIIIIICCCCC PPPPPRRRRROOOOOGGGGGRRRRREEEEESSSSSSSSSSIIIIIOOOOONNNNN (((((AAAAA.....PPPPP.....)))))
A sequence a , a ,a , ……, a is called an Arithmetic Progression (A.P.) when a – a = a – a = …..
1 2 3 n 2 1 3 2
= a – a . That means A. P. is a sequence in which each term is obtained by adding a constant d
n n–1
to the preceding term. This constant ‘d’ is called the common difference of the A.P. If 3 numbers a,
b, c are in A.P., we say
b – a = c – b or a + c = 2b; b is called the arithmetic mean between a and c.
EEEEExxxxxaaaaammmmmpppppllllleeeee::::: 11111))))) 2,5,8,11,14,17,…… is an A.P. in which d = 3 is the common diference.
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22222))))) 15,13,11,9,7,5,3,1,–1, is an A.P. in which –2 is the common difference.
SSSSSooooollllluuuuutttttiiiiiooooonnnnn::::: In (1) 2nd term = 5 , 1st term = 2, 3rd term = 8,
so 2nd term – 1st term = 5 – 2 = 3, 3rd term – 2nd term = 8 – 5 = 3
Here the difference between a term and the preceding term is same that is always constant. This
constant is called common difference.
Now in generel an A.P. series can be written as
a, a + d, a + 2d, a + 3d, a + 4d, ……
where ‘a’ is the 1st term and ‘d’ is the common difference.
Thus 1st term ( t ) = a = a + ( 1 – 1 ) d
1
2nd term ( t ) = a + d = a + ( 2 – 1 ) d
2
3rd term ( t ) = a + 2d = a + ( 3 – 1 ) d
3
4th term ( t ) = a + 3d = a + ( 4 – 1 ) d
4
…………………………………………….
nth term ( t ) = a + ( n – 1 ) d, where n is the position no. of the term .
n
Using this formula we can get
50th term (= t ) = a+ ( 50 – 1 ) d = a + 49d
50
EEEEExxxxxaaaaammmmmpppppllllleeeee 11111::::: Find the 7th term of the A.P. 8, 5, 2, –1, –4,…..
SSSSSooooollllluuuuutttttiiiiiooooonnnnn ::::: Here a = 8, d = 5 – 8 = –3
Now t = 8 + ( 7 – 1 ) d
7
= 8 + ( 7 – 1 ) (– 3 )
= 8 + 6 (– 3 )
= 8 – 18
= – 10
3 4 5 17
EEEEExxxxxaaaaammmmmpppppllllleeeee 22222 ::::: Which term of the AP , , ............is ?
7 7 7 7
3 4 3 1 17
SSSSSooooollllluuuuutttttiiiiiooooonnnnn ::::: a = , d= - = , t n =
7 7 7 7 7
We may write
17 3 1
= +(n-1) ´
7 7 7
or, 17 = 3 + ( n – 1)
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or, n = 17 – 2 = 15
17
.
Hence, 15th term of the A.P. is
7
EEEEExxxxxaaaaammmmmpppppllllleeeee 33333::::: If 5th and 12th terms of an A.P. are 14 and 35 respectively, find the A.P.
SSSSSooooollllluuuuutttttiiiiiooooonnnnn::::: Let a be the 1st term & d be the common difference of A.P.
t = a + 4d = 14
5
t = a + 11d = 35
12
On solving the above two equations:
7d = 21 = i.e., d = 3
and a = 14 – (4 × 3) = 14 – 12 = 2
Hence, the required A.P. is 2, 5, 8, 11, 14,……………
EEEEExxxxxaaaaammmmmpppppllllleeeee 44444::::: Divide 69 into three parts which are in A.P. and are such that the product of the 1st
two parts is 483.
SSSSSooooollllluuuuutttttiiiiiooooonnnnn::::: Given that the three parts are in A.P., let the three parts which are in A.P. be a – d, a,
a + d.........
Thus a – d + a + a + d = 69
or 3a = 69
or a = 23
So the three parts are 23 – d, 23, 23 + d
Since the product of first two parts is 483, therefore, we have
23 ( 23 – d ) = 483
or 23 – d = 483 / 23 = 21
or d = 23 – 21 = 2
Hence, the three parts which are in A.P. are
23 – 2 = 21, 23, 23 + 2 = 25
Finally the parts are 21, 23, 25.
EEEEExxxxxaaaaammmmmpppppllllleeeee 55555: Find the arithmetic mean between 4 and 10.
SSSSSooooollllluuuuutttttiiiiiooooonnnnn::::: We know that the A.M. of a & b is = ( a + b ) /2
Hence, The A. M between 4 & 10 = ( 4 + 10 ) /2 = 7
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EEEEExxxxxaaaaammmmmpppppllllleeeee 66666::::: Insert 4 arithmetic means between 4 and 324.
4, –, –, –, –, 324
SSSSSooooollllluuuuutttttiiiiiooooonnnnn::::: Here a= 4, d = ? n = 2 + 4 = 6, t = 324
n
Now t = a + ( n – 1 ) d
n
or 324= 4 + ( 6 – 1 ) d
or 320= 5d i.e., = i.e., d = 320 / 5 = 64
So the 1st AM = 4 + 64 = 68
2nd AM = 68 + 64 = 132
3rd AM = 132 + 64 = 196
4th AM = 196 + 64 = 260
SSSSSuuuuummmmm ooooofffff ttttthhhhheeeee fffffiiiiirrrrrsssssttttt nnnnn ttttteeeeerrrrrmmmmmsssss
Let S be the Sum, a be the 1st term and (cid:1) the last term of an A.P. If the number of term are n, then
(cid:1)
t = . Let d be the common difference of the A.P.
n
(cid:1) (cid:1) (cid:1)
Now S = a + ( a + d ) + ( a + 2d ) + .. + ( – 2d ) + ( – d ) +
(cid:1) (cid:1) (cid:1)
Again S = + ( – d ) + ( – 2d ) + …. + ( a + 2d ) + ( a + d ) + a
On adding the above, we have
(cid:1) (cid:1) (cid:1) (cid:1)
2S = ( a + ) + ( a + ) + ( a + ) + …… + ( a + )
(cid:1)
= n( a + )
(cid:1)
or S = n( a + ) / 2
NNNNNooooottttteeeee::::: The above formula may be used to determine the sum of n terms of an A.P. when the first
term a and the last term is given.
Now l = t = a + ( n – 1 ) d
n
n{a+a+(n-1)d}
∴ S=
2
n
or s= {2a+(n-1)d}
2
NNNNNooooottttteeeee::::: The above formula may be used when the first term a, common difference d and the number
of terms of an A.P. are given.
SSSSSuuuuummmmm ooooofffff 11111sssssttttt nnnnn nnnnnaaaaatttttuuuuurrrrraaaaalllll ooooorrrrr cccccooooouuuuunnnnntttttiiiiinnnnnggggg nnnnnuuuuummmmmbbbbbeeeeerrrrrsssss
S = 1 + 2 + 3 + ……. +……. ( n – 2 ) + ( n – 1 ) + n
Again S = n + ( n – 1 ) + ( n – 2 ) + ……… + 3 + 2 + 1
On adding the above, we get
2S = ( n + 1 ) + ( n + 1 ) +....... to n terms
or 2S = n ( n + 1 )
S = n( n + 1 )/2
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Then Sum of 1st, n natural number is n( n + 1 ) / 2
n(n+1)
i.e. 1 + 2 + 3 + ........ + n = .
2
SSSSSuuuuummmmm ooooofffff 11111sssssttttt nnnnn ooooodddddddddd nnnnnuuuuummmmmbbbbbeeeeerrrrr
S = 1 + 3 + 5 + …… + ( 2n – 1 )
Sum of 1st n odd number
S = 1 + 3 + 5 + …… + ( 2n – 1 )
Since S = n{ 2a + ( n –1 ) d } / 2, we find
n n
S = { 2.1 + ( n – 1 ) 2 } = ( 2n ) n2
2 2
or S = n2
Then sum of 1st, n odd numbers is n2, i.e. 1 + 3 + 5 + ..... + ( 2n – 1 ) = n2
SSSSSuuuuummmmm ooooofffff ttttthhhhheeeee SSSSSqqqqquuuuuaaaaarrrrreeeeesssss ooooofffff ttttthhhhheeeee 11111sssssttttt,,,,, nnnnn nnnnnaaaaatttttuuuuurrrrraaaaalllll nnnnnooooosssss.....
Let S = 12 + 22 + 32 + …… + n2
We know m3 – ( m – 1 ) 3 = 3m2 – 3m + 1
We put m = 1, 2, 3,……,n
13 – 0 = 3.12 – 3.1 + 1
23 – 13 = 3.22 – 3.2 + 1
33 – 23 = 3.32 – 3.3 + 1
…………………………..
+ n3 – ( n – 1 ) 3 = 3n2 – 3.n + 1
Adding both sides term by term,
n3 = 3S – 3 n ( n + 1 ) / 2 + n
or 2n3 = 6S – 3n2 – 3n + 2n
or 6S = 2n3 + 3n2 + n
or 6S = n ( 2n2 + 3n + 1 )
or 6S = n ( n + 1 ) ( 2n + 1 )
S = n( n + 1 )( 2n + 1 ) / 6
n(n+1) (2n+1)
TTTTThhhhhuuuuusssss sssssuuuuummmmm ooooofffff ttttthhhhheeeee sssssqqqqquuuuuaaaaarrrrreeeeesssss ooooofffff ttttthhhhheeeee 11111sssssttttt,,,,, nnnnn nnnnnaaaaatttttuuuuurrrrraaaaalllll nnnnnuuuuummmmmbbbbbeeeeerrrrrsssss iiiiisssss
6
n(n+1) (2n+1)
i.e. 12 + 22 + 32 + ........ + n2 = .
6 2
n(n+1)
Similarly, sum of the cubes of 1st n natural number can be found out as by taking
2
the identity
(cid:6)(cid:17)(cid:15)(cid:18)(cid:16) (cid:1)(cid:2)(cid:22)
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(cid:1)(cid:2)(cid:3)(cid:4)(cid:2)(cid:5)(cid:6)(cid:2)(cid:7)(cid:8)(cid:5)(cid:9)(cid:7)(cid:1)(cid:2)(cid:10)(cid:11)(cid:2)(cid:1)(cid:12)(cid:8)(cid:10)(cid:11)(cid:13)(cid:14)(cid:15)(cid:2)(cid:13)(cid:11)(cid:6)(cid:7)(cid:8)(cid:5)(cid:9)(cid:7)(cid:16)(cid:2)(cid:17)(cid:15)(cid:2)(cid:13)(cid:10)(cid:11)(cid:6)(cid:7)(cid:18)(cid:10)(cid:17)(cid:16)(cid:10)(cid:2)(cid:1)(cid:1)(cid:11)(cid:17)(cid:5)(cid:1)
m4 – ( m – 1 ) 4 = 4m3 – 6m2 + 4m – 1 and putting m = 1, 2, 3,…., n.
Thus
2
n(n+1)
13 + 23 + 33 + …. + n3 =
2
EEEEExxxxxeeeeerrrrrccccciiiiissssseeeee 66666 (((((AAAAA)))))
CCCCChhhhhoooooooooossssseeeee ttttthhhhheeeee mmmmmooooosssssttttt aaaaapppppppppprrrrroooooppppprrrrriiiiiaaaaattttteeeee oooooppppptttttiiiiiooooonnnnn ((((( aaaaa ))))),,,,, ((((( bbbbb ))))) ,,,,, ((((( ccccc ))))) ooooorrrrr (((((ddddd)))))
1. The nth element of the sequence 1, 3, 5, 7,….…..Is
(a) n (b) 2n – 1 (c) 2n +1 (d) none of these
2. The nth element of the sequence –1, 2, –4, 8 ….. is
(a) ( –1 )n2 n–1 (b) 2 n–1 (c) 2n (d) none of these
7
∑
3. 2i-1 can be written as
i=4
(a) 7+ 9+ 11+ 13 (b) 2 7+ 2 9+ 2 11+2 13
(c) 2 7+ 2 9+ 2 11+2 13 (d) none of these.
4. –5, 25, –125 , 625, ….. can be written as
∝ ∝ ∝
(a)
∑ (-5)k
(b)
∑ 5k
(c)
∑ − 5k
(d) none of these
k=1 k=1 k=1
5. The first three terms of sequence when nth term t is n2 – 2n are
n
(a) –1, 0, 3 (b) 1, 0, 2 (c) –1, 0, –3 (d) none of these
6. Which term of the progression –1, –3, –5, …. Is –39
(a) 21st (b) 20th (c) 19th (d) none of these
7. The value of x such that 8x + 4, 6x – 2, 2x + 7 will form an AP is
(a) 15 (b) 2 (c) 15/2 (d) none of the these
8. The mth term of an A. P. is n and nth term is m. The r th term of it is
(a) m + n +r (b) n + m – 2r (c) m + n + r/2 (d) m + n – r
2 1
9. The number of the terms of the series 10 + 9 + 9 + 9 + ............will amount to 155 is
3 3
(a) 30 (b) 31 (c) 32 (d) none of these
10. The nth term of the series whose sum to n terms is 5n2 + 2n is
(a) 3n – 10 (b) 10n – 2 (c) 10n – 3 (d) none of these
11. The 20th term of the progression 1, 4, 7, 10.................is
(a) 58 (b) 52 (c) 50 (d) none of these
12. The last term of the series 5, 7, 9,….. to 21 terms is
(a) 44 (b) 43 (c) 45 (d) none of these
(cid:1)(cid:2)(cid:23) (cid:4)(cid:5)(cid:6)(cid:6)(cid:5)(cid:7)(cid:8) (cid:9)(cid:10)(cid:5)(cid:11)(cid:12)(cid:4)(cid:12)(cid:13)(cid:7)(cid:4)(cid:14)(cid:8) (cid:15)(cid:13)(cid:16)(cid:15)
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13. The last term of the A.P. 0.6, 1.2, 1.8,… to 13 terms is
(a) 8.7 (b) 7.8 (c) 7.7 (d) none of these
14. The sum of the series 9, 5, 1,…. to 100 terms is
(a) –18900 (b) 18900 (c) 19900 (d) none of these
15. The two arithmetic means between –6 and 14 is
1 1
(a) 2/3, 1/3 (b) 2/3, 7 (c) – 2/3, – 7 (d) none of these
3 3
16. The sum of three integers in AP is 15 and their product is 80. The integers are
(a) 2, 8, 5 (b) 8, 2, 5 (c) 2, 5, 8 (d) 8, 5, 2
17. The sum of n terms of an AP is 3n2 + 5n. The series is
(a) 8, 14, 20, 26 (b) 8, 22, 42, 68 (c) 22, 68, 114, .... (d) none of these
18. The number of numbers between 74 and 25556 divisible by 5 is
(a) 5090 (b) 5097 (c) 5095 (d) none of these
19. The pth term of an AP is (3p – 1)/6. The sum of the first n terms of the AP is
(a) n (3n + 1) (b) n/12 (3n + 1) (c) n/12 (3n – 1) (d) none of these
20. The arithmetic mean between 33 and 77 is
(a) 50 (b) 45 (c) 55 (d) none of these
21. The 4 arithmetic means between –2 and 23 are
(a) 3, 13, 8, 18 (b) 18, 3, 8, 13 (c) 3, 8, 13, 18 (d) none of these
22. The first term of an A.P is 14 and the sums of the first five terms and the first ten terms are
equal is magnitude but opposite in sign. The 3rd term of the AP is
4
(a) 6 (b) 6 (c) 4/11 (d) none of these
11
23. The sum of a certain number of terms of an AP series –8, –6, –4, …… is 52. The number of
terms is
(a) 12 (b) 13 (c) 11 (d) none of these
24. The 1st and the last term of an AP are –4 and 146. The sum of the terms is 7171. The number
of terms is
(a) 101 (b) 100 (c) 99 (d) none of these
25. The sum of the series 3 ½ + 7 + 10 ½ + 14 + …. To 17 terms is
(a) 530 (b) 535 (c) 535 ½ (d) none of these
66666.....44444 GGGGGEEEEEOOOOOMMMMMEEEEETTTTTRRRRRIIIIICCCCC PPPPPRRRRROOOOOGGGGGRRRRREEEEESSSSSSSSSSIIIIIOOOOONNNNN (((((GGGGG.....PPPPP.....)))))
If in a sequence of terms each term is constant multiple of the proceeding term, then the sequence
is called a Geometric Progression (G.P). The constant multiplier is called the common ratio
EEEEExxxxxaaaaammmmmpppppllllleeeeesssss::::: 1) In 5, 15, 45, 135,….. common ratio is 15/5 = 3
2) In 1, 1/ , 1/ , 1/, … common ratio is (½) /1=½
2 4 8
3) In 2, –6, 18, –54, …. common ratio is (–6) / 2 =–3
(cid:6)(cid:17)(cid:15)(cid:18)(cid:16) (cid:1)(cid:2)(cid:24)
Copyright -The Institute of Chartered Accountants of India
(cid:1)(cid:2)(cid:3)(cid:4)(cid:2)(cid:5)(cid:6)(cid:2)(cid:7)(cid:8)(cid:5)(cid:9)(cid:7)(cid:1)(cid:2)(cid:10)(cid:11)(cid:2)(cid:1)(cid:12)(cid:8)(cid:10)(cid:11)(cid:13)(cid:14)(cid:15)(cid:2)(cid:13)(cid:11)(cid:6)(cid:7)(cid:8)(cid:5)(cid:9)(cid:7)(cid:16)(cid:2)(cid:17)(cid:15)(cid:2)(cid:13)(cid:10)(cid:11)(cid:6)(cid:7)(cid:18)(cid:10)(cid:17)(cid:16)(cid:10)(cid:2)(cid:1)(cid:1)(cid:11)(cid:17)(cid:5)(cid:1)
IIIIIlllllllllluuuuussssstttttrrrrraaaaatttttiiiiiooooonnnnnsssss::::: Consider the following series :–
(i) 1 + 4 + 16 + 64 + …………….
Here second term / 1st term = 4/1 = 4; third term / second term = 16/4 = 4
fourth term/third term = 64/16 = 4 and so on.
Thus, we find that, in the entire series, the ratio of any term and the term preceding it, is a
constant.
(ii) 1/3 – 1/9 + 1/27 – 1/81 + ………….
Here second term / 1st term = (–1/9) / ( 1/3) = –1/3
third term / second term = ( 1/27 ) / ( –1/9 ) = –1/3
fourth term / third term = ( –1/81 ) / (1/27 ) = –1/3 and so on.
Here also, in the entire series, the ratio of any term and the term preceding one is constant.
The above mentioned series are known as GGGGGeeeeeooooommmmmeeeeetttttrrrrriiiiiccccc SSSSSeeeeerrrrriiiiieeeeesssss.....
Let us consider the sequence a, ar, ar2, ar3, ….
1st term = a, 2nd term = ar = ar 2–1, 3rd term = ar2 = ar3–1, 4th term = ar3 = ar 4 –1, …..
Similarly nth term of GP t = ar n–1
n
Anyterm t
= n
Thus, common ratio =
Precedingterm t
n-1
= ar n–1/ar n–2 = r
Thus, general term of a G.P is given by ar n–1 and the general form of G.P. is
a + ar + ar2 + ar3 +……. ….
t ar
2 =
For example, r =
t a
1
t t t
2 = 3 = 4 =....
So r =
t t t
1 2 3
EEEEExxxxxaaaaammmmmpppppllllleeeee 11111::::: If a, ar, ar2, ar3, …. be in G.P. Find the common ratio.
SSSSSooooollllluuuuutttttiiiiiooooonnnnn::::: 1st term = a, 2nd term = ar
Ratio of any term to its preceding term = ar/a = r = common ratio.
EEEEExxxxxaaaaammmmmpppppllllleeeee 22222::::: Which term of the progression 1, 2, 4, 8,… is 256?
SSSSSooooollllluuuuutttttiiiiiooooonnnnn ::::: a = 1, r = 2/1 = 2, n = ? t = 256
n
t = ar n–1
n
or 256 = 1 × 2 n–1 i.e., 28 = 2 n–1 or, n – 1 = 8 i.e., n = 9
(cid:1)(cid:2)(cid:25)(cid:26) (cid:4)(cid:5)(cid:6)(cid:6)(cid:5)(cid:7)(cid:8) (cid:9)(cid:10)(cid:5)(cid:11)(cid:12)(cid:4)(cid:12)(cid:13)(cid:7)(cid:4)(cid:14)(cid:8) (cid:15)(cid:13)(cid:16)(cid:15)
Copyright -The Institute of Chartered Accountants of India
Thus 9th term of the G. P. is 256
66666.....55555 GGGGGEEEEEOOOOOMMMMMEEEEETTTTTRRRRRIIIIICCCCC MMMMMEEEEEAAAAANNNNN
If a, b, c are in G.P we get b/a = c/b => b2 = ac, b is called the geometric mean between a
and c
EEEEExxxxxaaaaammmmmpppppllllleeeee 11111::::: Insert 3 geometric means between 1/9 and 9.
SSSSSooooollllluuuuutttttiiiiiooooonnnnn::::: 1/9, –, –, –, 9
a = 1/9, r = ?, n = 2 + 3 = 5, t = 9
n
we know t = ar n–1
n
or 1/9 × r 5–1 = 9
or r4 = 81 = 34 => r = 3
Thus 1st G. M = 1/9 × 3 = 1/3
2nd G. M = 1/3 × 3 = 1
3rd G. M = 1× 3 = 3
EEEEExxxxxaaaaammmmmpppppllllleeeee 22222::::: Find the G.P where 4th term is 8 and 8th term is 128/625
SSSSSooooollllluuuuutttttiiiiiooooonnnnn ::::: Let a be the 1st term and r be the common ratio.
By the question t = 8 and t = 128/625
4 8
So ar3 = 8 and ar7 = 128 / 625
128
Therefore ar7 / ar3 = => r4 = 16 / 625 =( +2/5 )4 => r = 2/5 and –2 /5
625´ 8
Now ar3 = 8 => a × (2/5) 3 = 8 => a = 125
Thus the G. P is
125, 50, 20, 8, 16/5, ………..
When r = –2/5 , a = –125 and the G.P is –125, 50, –20, 8, –16/5 ,………
Finally, the G.P. is 125, 50, 20, 8, 16/5, ………..
or, –125, 50, –20, 8, –16/5,………
SSSSSuuuuummmmm ooooofffff fffffiiiiirrrrrsssssttttt nnnnn ttttteeeeerrrrrmmmmmsssss ooooofffff aaaaa GGGGG PPPPP
Let a be the 1st term and r be the common ratio. So the 1st n terms are a, ar, ar2, …... ar n–1.
If S be the sum of n terms,
S = a + ar + ar2 + ……+ ar n–1 (i)
...........................................
n
Now rS = ar + ar2 + ….. + ar n–1 + arn (ii)
.....................................
n
(cid:6)(cid:17)(cid:15)(cid:18)(cid:16) (cid:1)(cid:2)(cid:25)(cid:25)
Copyright -The Institute of Chartered Accountants of India
(cid:1)(cid:2)(cid:3)(cid:4)(cid:2)(cid:5)(cid:6)(cid:2)(cid:7)(cid:8)(cid:5)(cid:9)(cid:7)(cid:1)(cid:2)(cid:10)(cid:11)(cid:2)(cid:1)(cid:12)(cid:8)(cid:10)(cid:11)(cid:13)(cid:14)(cid:15)(cid:2)(cid:13)(cid:11)(cid:6)(cid:7)(cid:8)(cid:5)(cid:9)(cid:7)(cid:16)(cid:2)(cid:17)(cid:15)(cid:2)(cid:13)(cid:10)(cid:11)(cid:6)(cid:7)(cid:18)(cid:10)(cid:17)(cid:16)(cid:10)(cid:2)(cid:1)(cid:1)(cid:11)(cid:17)(cid:5)(cid:1)
Subtracting (i) from (ii)
S – rS = a – ar n
n n
or S (1 – r) = a (1 – rn)
n
or S = a ( 1 – rn) / ( 1 – r ) when r < 1
n
S = a ( rn – 1 ) / ( r – 1 ) when r > 1
n
If r = 1 , then S = a + a + a+ ……….. to n terms
n
= na
If the nth term of the G. P be l then (cid:1) = arn–1
lr-a
Therefore, S = (arn –a ) / (r – 1) = (a rn –1 r –a) / (r – 1) =
n r-1
So, when the last term of the G. P is known, we use this formula.
SSSSSuuuuummmmm ooooofffff iiiiinnnnnfffffiiiiinnnnniiiiittttteeeee gggggeeeeeooooommmmmeeeeetttttrrrrriiiiiccccc ssssseeeeerrrrriiiiieeeeesssss
S = a ( 1 – rn ) / (1 – r) when r < 1
= a (1 – 1/Rn) / ( 1 – 1/R ) (since r < 1 , we take r = 1/R).
If n →→→→→ ∝ , 1/Rn →→→→→ 0
a
Thus S = , r<1
µ
1–r
a
i.e. Sum of G.P. upto infinity is , where r < 1
1–r
a
Also, S = , if -1<r<1.
µ
1–r
EEEEExxxxxaaaaammmmmpppppllllleeeee 11111::::: Find the sum of 1 + 2 + 4 + 8 + … to 8 terms.,
SSSSSooooollllluuuuutttttiiiiiooooonnnnn::::: Here a = 1, r = 2/1 = 2 , n = 8
Let S = 1 + 2 + 4 + 8 + …… to 8 terms
= 1 ( 28 – 1 ) / ( 2 – 1 ) = 28 – 1 = 255
EEEEExxxxxaaaaammmmmpppppllllleeeee 22222::::: Find the sum to n terms of 6 + 27 + 128 + 629 + …….
SSSSSooooollllluuuuutttttiiiiiooooonnnnn::::: Required Sum= ( 5 + 1 ) + ) (52 + 2 ) + ( 53 +3 ) ( 54 + 4 ) + … to n terms
= ( 5 + 52 +53 + …… + 5n ) + ( 1 + 2 + 3 + .. + n terms)
= {5 ( 5n – 1 ) / (5 – 1 )} + {n ( n + 1 ) / 2}
= {5 ( 5n – 1 ) /4} + {n ( n + 1 ) / 2}
EEEEExxxxxaaaaammmmmpppppllllleeeee 33333::::: Find the sum to n terms of the series
3 + 33 + 333 + …….
(cid:1)(cid:2)(cid:25)(cid:3) (cid:4)(cid:5)(cid:6)(cid:6)(cid:5)(cid:7)(cid:8) (cid:9)(cid:10)(cid:5)(cid:11)(cid:12)(cid:4)(cid:12)(cid:13)(cid:7)(cid:4)(cid:14)(cid:8) (cid:15)(cid:13)(cid:16)(cid:15)
Copyright -The Institute of Chartered Accountants of India
SSSSSooooollllluuuuutttttiiiiiooooonnnnn::::: Let S denote the required sum.
i.e. S = 3 + 33 + 333 + ………….. to n terms
= 3 (1 + 11 + 111 + ……. to n terms)
3
= (9 + 99 + 999 + …. to n terms)
9
3
= {( 10 – 1 ) + ( 102 – 1 ) + ( 103 – 1 ) + … + ( 10n – 1 )}
9
3
= {( 10 + 102 + 103 + …. + 10n ) – n}
9
3
= {10 ( 1 + 10 + 102 + … + 10 n–1 ) – n}
9
3
= [{10 ( 10n – 1 ) / (10 – 1)} – n]
9
3
= (10 n+1 – 10 – 9n)
81
1
= (10 n+1 – 9n – 10)
27
EEEEExxxxxaaaaammmmmpppppllllleeeee 44444::::: Find the sum of n terms of the series 0.7 + 0.77 + 0.777 + …. to n terms
SSSSSooooollllluuuuutttttiiiiiooooonnnnn ::::: Let S denote the required sum.
i.e. S = 0.7 + 0.77 + 0.777 + ….. to n terms
= 7 (0.1 + 0.11 + 0.111 + …. to n terms)
7
= (0.9 + 0.99 + 0.999 + … to n terms )
9
7
= {(1 – 1/10 ) + ( 1 – 1/102 ) + ( 1 – 1/103 ) + … + ( 1 – 1/ 10n )}
9
7 1
= {n – ( 1 + 1/10 + 1/102 + …. + 1/10 n–1)}
9 10
7 1
So S = {n – ( 1 – 1/10n )/(1 – 1/10 ) }
9 10
7
= {n – ( 1 – 10 –n ) / 9 ) }
9
7
= {9n – 1 + 10 –n }
81
(cid:2)(cid:2)
EEEEExxxxxaaaaammmmmpppppllllleeeee 55555::::: Evaluate 0.2175 using the sum of an infinite geometric series.
(cid:6)(cid:17)(cid:15)(cid:18)(cid:16) (cid:1)(cid:2)(cid:25)(cid:19)
Copyright -The Institute of Chartered Accountants of India
(cid:1)(cid:2)(cid:3)(cid:4)(cid:2)(cid:5)(cid:6)(cid:2)(cid:7)(cid:8)(cid:5)(cid:9)(cid:7)(cid:1)(cid:2)(cid:10)(cid:11)(cid:2)(cid:1)(cid:12)(cid:8)(cid:10)(cid:11)(cid:13)(cid:14)(cid:15)(cid:2)(cid:13)(cid:11)(cid:6)(cid:7)(cid:8)(cid:5)(cid:9)(cid:7)(cid:16)(cid:2)(cid:17)(cid:15)(cid:2)(cid:13)(cid:10)(cid:11)(cid:6)(cid:7)(cid:18)(cid:10)(cid:17)(cid:16)(cid:10)(cid:2)(cid:1)(cid:1)(cid:11)(cid:17)(cid:5)(cid:1)
(cid:2)(cid:2)
SSSSSooooollllluuuuutttttiiiiiooooonnnnn::::: 0.2175 = 0.2175757575 …….
0.217 & 5 &= 0.21 + 0.0075 + 0.000075 + ….
= 0.21 + 75 ( 1 + 1/102 + 1/104 + …. ) / 104
= 0.21 + 75 {1 / (1– 1/102} / 104
= 0.21 + (75/104) × 102 /99
=21/100 + (¾ ) × (1/99 )
= 21/100 + 1/132
= ( 693 + 25 )/3300 = 718/3300 = 359/1650
EEEEExxxxxaaaaammmmmpppppllllleeeee 66666::::: Find three numbers in G. P whose sum is 19 and product is 216.
SSSSSooooollllluuuuutttttiiiiiooooonnnnn::::: Let the 3 numbers be a/r, a, ar.
According to the question a/r × a × ar = 216
ora3 = 63 => a =6
So the numbers are 6/r, 6, 6r
Again 6/r + 6 + 6r = 19
or 6/r + 6r = 13
or 6 + 6r2 = 13r
or 6r2 – 13r + 6 = 0
or 6r2 – 4r – 9r + 6 = 0
or 2r(3r –2) – 3 (3r – 2) = 2
or (3r – 2) (2r – 3) = 0 or, r = 2/3 , 3/2
So the numbers are
6/(2/3), 6, 6 × (2/3 ) = 9 , 6 , 4
or 6/(3/2), 6 , 6 × (3/2) = 4 , 6 , 9
EEEEExxxxxeeeeerrrrrccccciiiiissssseeeee 66666 (((((BBBBB)))))
CCCCChhhhhoooooooooossssseeeee ttttthhhhheeeee mmmmmooooosssssttttt aaaaapppppppppprrrrroooooppppprrrrriiiiiaaaaattttteeeee oooooppppptttttiiiiiooooonnnnn (((((aaaaa))))),,,,, (((((bbbbb))))),,,,, (((((ccccc))))) ooooorrrrr (((((ddddd)))))
1. The 7th term of the series 6, 12, 24,……is
(a) 384 (b) 834 (c) 438 (d) none of these
2. t of the series 6, 12, 24,…is
8
(a) 786 (b) 768 (c) 867 (c) none of these
3. t of the series –128, 64, –32, ….is
12
(a) – 1/16 (b) 16 (c) 1/16 (d) none of these
4. The 4th term of the series 0.04, 0.2, 1, … is
(a) 0.5 (b) ½ (c) 5 (d) none of these
(cid:1)(cid:2)(cid:25)(cid:20) (cid:4)(cid:5)(cid:6)(cid:6)(cid:5)(cid:7)(cid:8) (cid:9)(cid:10)(cid:5)(cid:11)(cid:12)(cid:4)(cid:12)(cid:13)(cid:7)(cid:4)(cid:14)(cid:8) (cid:15)(cid:13)(cid:16)(cid:15)
Copyright -The Institute of Chartered Accountants of India
5. The last term of the series 1, 2, 4,…. to 10 terms is
(a) 512 (b) 256 (c) 1024 (d) none of these
6. The last term of the series 1, –3, 9, –27 up to 7 terms is
(a) 297 (b) 729 (c) 927 (d) none of these
7. The last term of the series x2, x, 1, …. to 31 terms is
(a) x28 (b) 1/x (c) 1/x28 (d) none of these
8. The sum of the series –2, 6, –18, …. to 7 terms is
(a) –1094 (b) 1094 (c) – 1049 (d) none of these
9. The sum of the series 24, 3, 8, 1, 2, 7,… to 8 terms is
13 1
(a) 36 (b) 36 (c) 36 (d) none of these
30 9
1 3
10. The sum of the series +1+ +……to 18 terms is
3 3
(cid:1)(cid:2) + (cid:3)(cid:4) 9841
(a) 9841 (b) 9841 (c) (d) none of these
(cid:3) 3
11. The second term of a G P is 24 and the fifth term is 81. The series is
(a) 16, 36, 24, 54,.. (b) 24, 36, 53,… (c) 16, 24, 36, 54,.. (d) none of these
12. The sum of 3 numbers of a G P is 39 and their product is 729. The numbers are
(a) 3, 27, 9 (b) 9, 3, 27 (c) 3, 9, 27 (d) none of these
13. In a G. P, the product of the first three terms 27/8. The middle term is
(a) 3/2 (b) 2/3 (c) 2/5 (d) none of these
14. If you save 1 paise today, 2 paise the next day 4 paise the succeeding day and so on, then
your total savings in two weeks will be
(a) Rs. 163 (b) Rs. 183 (c) Rs. 163.83 (d) none of these
15. Sum of n terms of the series 4 + 44 + 444 + … is
(a) 4/9 { 10/9 ( 10n –1 ) –n } (b) 10/9 ( 10n –1 ) –n
(c) 4/9 ( 10n –1 ) –n (d) none of these
16. Sum of n terms of the series 0.1 + 0.11 + 0.111 + … is
(a) 1/9 {n – ( 1– ( 0.1 )n )} (b) 1/9 {n – (1–(0.1)n)/9}
(c) n– 1 – (0.1)n/9 (d) none of these
17. The sum of the first 20 terms of a G. P is 244 times the sum of its first 10 terms. The common
ratio is
(a) ± 3 (b) ±3 (c) 3 (d) none of these
18. Sum of the series 1 + 3 + 9 + 27 +….is 364. The number of terms is
(a) 5 (b) 6 (c) 11 (d) none of these
(cid:6)(cid:17)(cid:15)(cid:18)(cid:16) (cid:1)(cid:2)(cid:25)(cid:21)
Copyright -The Institute of Chartered Accountants of India
(cid:1)(cid:2)(cid:3)(cid:4)(cid:2)(cid:5)(cid:6)(cid:2)(cid:7)(cid:8)(cid:5)(cid:9)(cid:7)(cid:1)(cid:2)(cid:10)(cid:11)(cid:2)(cid:1)(cid:12)(cid:8)(cid:10)(cid:11)(cid:13)(cid:14)(cid:15)(cid:2)(cid:13)(cid:11)(cid:6)(cid:7)(cid:8)(cid:5)(cid:9)(cid:7)(cid:16)(cid:2)(cid:17)(cid:15)(cid:2)(cid:13)(cid:10)(cid:11)(cid:6)(cid:7)(cid:18)(cid:10)(cid:17)(cid:16)(cid:10)(cid:2)(cid:1)(cid:1)(cid:11)(cid:17)(cid:5)(cid:1)
19. The product of 3 numbers in G P is 729 and the sum of squares is 819. The numbers are
(a) 9, 3, 27 (b) 27, 3, 9 (c) 3, 9, 27 (d) none of these
20. The sum of the series 1 + 2 + 4 + 8 + .. to n term
(a) 2n –1 (b) 2n – 1 (c) 1/2n – 1 (d) none of these
21. The sum of the infinite GP 14, – 2, + 2/7, – 2/49, + … is
1 1
(a) 4 (b) 12 (c) 12 (d) none of these
12 4
22. The sum of the infinite G. P. 1 - 1/3 + 1/9 - 1/27 +... is
(a) 0.33 (b) 0.57 (c) 0.75 (d) none of these
23. The number of terms to be taken so that 1 + 2 + 4 + 8 + will be 8191 is
(a) 10 (b) 13 (c) 12 (d) none of these
24. Four geometric means between 4 and 972 are
(a) 12,30,100,324 (b) 12,24,108,320 (c) 10,36,108,320 (d) none of these
IIIIIlllllllllluuuuussssstttttrrrrraaaaatttttiiiiiooooonnnnnsssss :::::
(((((IIIII))))) A person is employed in a company at Rs. 3000 per month and he would get an increase of
Rs. 100 per year. Find the total amount which he receives in 25 years and the monthly salary
in the last year.
SSSSSooooollllluuuuutttttiiiiiooooonnnnn:::::
He gets in the 1st year at the Rate of 3000 per month;
In the 2nd year he gets at the rate of Rs. 3100 per month;
In the 3rd year at the rate of Rs. 3200 per month so on.
In the last year the monthly salary will be
Rs. {3000 + ( 25 – 1 ) × 100} = Rs. 5400
n
Total amount = Rs. 12 ( 3000 + 3100 + 3200 +… + 5400) Use S
n
= (a+l)
2
= Rs. 12 × 25/2 (3000 + 5400)
= Rs. 150 × 8400
= Rs. 12,60,000
(((((IIIIIIIIII))))) A person borrows Rs. 8,000 at 2.76% Simple Interest per annum. The principal and the interest
are to be paid in the 10 monthly instalments. If each instalment is double the preceding one,
find the value of the first and the last instalment.
SSSSSooooollllluuuuutttttiiiiiooooonnnnn:::::
Interest to be paid = 2.76 × 10 × 8000 / 100 × 12 = Rs. 184
Total amount to be paid in 10 monthly instalment is Rs. (8000 + 184) = Rs. 8184
The instalments form a G P with common ratio 2 and so Rs. 8184 = a (210 – 1 ) / ( 2 – 1 ),
a = 1st instalment
(cid:1)(cid:2)(cid:25)(cid:1) (cid:4)(cid:5)(cid:6)(cid:6)(cid:5)(cid:7)(cid:8) (cid:9)(cid:10)(cid:5)(cid:11)(cid:12)(cid:4)(cid:12)(cid:13)(cid:7)(cid:4)(cid:14)(cid:8) (cid:15)(cid:13)(cid:16)(cid:15)
Copyright -The Institute of Chartered Accountants of India
Here a = Rs. 8184 / 1023 = Rs. 8
The last instalment = ar 10—1 = 8 × 29 = 8 × 512 = Rs. 4096
EEEEExxxxxeeeeerrrrrccccciiiiissssseeeee 66666 (((((ccccc)))))
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1. Three numbers are in AP and their sum is 21. If 1, 5, 15 are added to them respectively,
they form a G. P. The numbers are
(a) 5, 7, 9 (b) 9, 5, 7 (c) 7, 5, 9 (d) none of these
2. The sum of 1 + 1/3 + 1/32 + 1/33 + … + 1/3 n –1 is
(a) 2/3 (b) 3/2 (c) 4/5 (d) none of these
3. The sum of the infinite series 1 + 2/3 + 4/9 + .. is
(a) 1/3 (b) 3 (c) 2/3 (d) none of these
4. The sum of the first two terms of a G.P. is 5/3 and the sum to infinity of the series is 3. The
common ratio is
(a) 1/3 (b) 2/3 (c) – 2/3 (d) none of these
5. If p, q and r are in A.P. and x, y, z are in G.P. then xq–r. y r–p. zp–q is equal to
(a) 0 (b) –1 (c) 1 (d) none of these
6. The sum of three numbers in G.P. is 70. If the two extremes by multiplied each by 4 and the
mean by 5, the products are in AP. The numbers are
(a) 12, 18, 40 (b) 10, 20, 40 (c) 40, 20, 10 (d) none of these
7. The sum of 3 numbers in A.P. is 15. If 1, 4 and 19 be added to them respectively, the results
are is G. P. The numbers are
(a) 26, 5, –16 (b) 2, 5, 8 (c) 5, 8, 2 (d) none of these
8. Given x, y, z are in G.P. and xp = yq = zσ, then 1/p , 1/q, 1/σ are in
(a) A.P. (b) G.P. (c) Both A.P. and G.P.(d) none of these
9. If the terms 2x, (x+10) and (3x+2) be in A.P., the value of x is
(a) 7 (b) 10 (c) 6 (d) none of these
10. If A be the A.M. of two positive unequal quantities x and y and G be their G. M, then
(a) A < G (b) A>G (c) A ≥ G (d) A ≤ G
11. The A.M. of two positive numbers is 40 and their G. M. is 24. The numbers are
(a) (72, 8) (b) (70, 10) (c) (60, 20) (d) none of these
12. Three numbers are in A.P. and their sum is 15. If 8, 6, 4 be added to them respectively, the
numbers are in G.P. The numbers are
(a) 2, 6, 7 (b) 4, 6, 5 (c) 3, 5, 7 (d) none of these
13. The sum of four numbers in G. P. is 60 and the A.M. of the 1st and the last is 18. The
numbers are
(a) 4, 8, 16, 32 (b) 4, 16, 8, 32 (c) 16, 8, 4, 20 (d) none of these
(cid:6)(cid:17)(cid:15)(cid:18)(cid:16) (cid:1)(cid:2)(cid:25)(cid:22)
Copyright -The Institute of Chartered Accountants of India
(cid:1)(cid:2)(cid:3)(cid:4)(cid:2)(cid:5)(cid:6)(cid:2)(cid:7)(cid:8)(cid:5)(cid:9)(cid:7)(cid:1)(cid:2)(cid:10)(cid:11)(cid:2)(cid:1)(cid:12)(cid:8)(cid:10)(cid:11)(cid:13)(cid:14)(cid:15)(cid:2)(cid:13)(cid:11)(cid:6)(cid:7)(cid:8)(cid:5)(cid:9)(cid:7)(cid:16)(cid:2)(cid:17)(cid:15)(cid:2)(cid:13)(cid:10)(cid:11)(cid:6)(cid:7)(cid:18)(cid:10)(cid:17)(cid:16)(cid:10)(cid:2)(cid:1)(cid:1)(cid:11)(cid:17)(cid:5)(cid:1)
14. A sum of Rs. 6240 is paid off in 30 instalments such that each instalment is Rs. 10 more
than the proceeding installment. The value of the 1st instalment is
(a) Rs. 36 (b) Rs. 30 (c) Rs. 60 (d) none of these
15. The sum of 1.03 + ( 1.03 ) 2 + ( 1.03 ) 3 + …. to n terms is
(a) 103 {(1.03)n – 1} (b) 103/3 {(1.03 )n – 1} (c) (1.03)n –1 (d) none of these
16. If x, y, z are in A.P. and x, y, (z + 1) are in G.P. then
(a) (x – z)2 = 4x (b) z2 = (x – y) (c) z = x – y (d) none of these
17. The numbers x, 8, y are in G.P. and the numbers x, y, –8 are in A.P. The value of x and y
are
(a) (–8, –8) (b) (16, 4) (c) (8, 8) (d) none of these
18. The nth term of the series 16, 8, 4,…. Is 1/217. The value of n is
(a) 20 (b) 21 (c) 22 (d) none of these
19. The sum of n terms of a G.P. whose first terms 1 and the common ratio is 1/ , is equal to
2
127
1 . The value of n is
128
(a) 7 (b) 8 (c) 6 (d) none of these
20. t of a G.P. in x, t = y and t = z. Then
4 10 16
(a) x2 = yz (b) z2 = xy (c) y2 = zx (d) none of these
21. If x, y, z are in G.P., then
(a) y2 = xz (b) y ( z2 + x2 ) = x ( z2 + y2 ) (c) 2y = x+z (d) none of these
22. The sum of all odd numbers between 200 and 300 is
(a) 11600 (b) 12490 (c) 12500 (d) 24750
23. The sum of all natural numbers between 500 and 1000 which are divisible by 13, is
(a) 28405 (b) 24805 (c) 28540 (d) none of these
24. If unity is added to the sum of any number of terms of the A.P. 3, 5, 7, 9,…... the resulting
sum is
(a) ‘a’ perfect cube (b) ‘a’ perfect square (c) ‘a’ number (d) none of these
25. The sum of all natural numbers from 100 to 300 which are exactly divisible by 4 or 5 is
(a) 10200 (b) 15200 (c) 16200 (d) none of these
26. The sum of all natural numbers from 100 to 300 which are exactly divisible by 4 and 5 is
(a) 2200 (b) 2000 (c) 2220 (d) none of these
27. A person pays Rs. 975 by monthly instalment each less then the former by Rs. 5. The first
instalment is Rs. 100. The time by which the entire amount will be paid is
(a) 10 months (b) 15 months (c) 14 months (d) none of these
(cid:1)(cid:2)(cid:25)(cid:23) (cid:4)(cid:5)(cid:6)(cid:6)(cid:5)(cid:7)(cid:8) (cid:9)(cid:10)(cid:5)(cid:11)(cid:12)(cid:4)(cid:12)(cid:13)(cid:7)(cid:4)(cid:14)(cid:8) (cid:15)(cid:13)(cid:16)(cid:15)
Copyright -The Institute of Chartered Accountants of India
28. A person saved Rs. 16,500 in ten years. In each year after the first year he saved Rs. 100
more than he did in the preceding year. The amount of money he saved in the 1st year was
(a) Rs. 1000 (b) Rs. 1500 (c) Rs. 1200 (d) none of these
29. At 10% C.I. p.a., a sum of money accumulate to Rs. 9625 in 5 years. The sum invested
initially is
(a) Rs. 5976.37 (b) Rs. 5970 (c) Rs. 5975 (d) Rs. 5370.96
30. The population of a country was 55 crose in 2005 and is growing at 2% p.a C.I. the
population is the year 2015 is estimated as
(a) 5705 (b) 6005 (c) 6700 (d) none of these
(cid:6)(cid:17)(cid:15)(cid:18)(cid:16) (cid:1)(cid:2)(cid:25)(cid:24)
Copyright -The Institute of Chartered Accountants of India
(cid:1)(cid:2)(cid:3)(cid:4)(cid:2)(cid:5)(cid:6)(cid:2)(cid:7)(cid:8)(cid:5)(cid:9)(cid:7)(cid:1)(cid:2)(cid:10)(cid:11)(cid:2)(cid:1)(cid:12)(cid:8)(cid:10)(cid:11)(cid:13)(cid:14)(cid:15)(cid:2)(cid:13)(cid:11)(cid:6)(cid:7)(cid:8)(cid:5)(cid:9)(cid:7)(cid:16)(cid:2)(cid:17)(cid:15)(cid:2)(cid:13)(cid:10)(cid:11)(cid:6)(cid:7)(cid:18)(cid:10)(cid:17)(cid:16)(cid:10)(cid:2)(cid:1)(cid:1)(cid:11)(cid:17)(cid:5)(cid:1)
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17. a 18. b 19. b 20. c 21. c 22. a 23. b 24. a
25. c
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17. a 18. b 19. c 20. a 21. b 22. c 23. b 24. d
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9. c 10. b 11. a 12. c 13. a 14. d 15. b 16. a
17. b 18. c 19. b 20. c 21. a 22. d 23. a 24. b
25. c 26. a 27. b 28. c 29. d 30. d
(cid:1)(cid:2)(cid:3)(cid:26) (cid:4)(cid:5)(cid:6)(cid:6)(cid:5)(cid:7)(cid:8) (cid:9)(cid:10)(cid:5)(cid:11)(cid:12)(cid:4)(cid:12)(cid:13)(cid:7)(cid:4)(cid:14)(cid:8) (cid:15)(cid:13)(cid:16)(cid:15)
Copyright -The Institute of Chartered Accountants of India
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1. If a, b, c are in A.P. as well as in G.P. then –
(A) They are also in H.P. (Harmonic Progression) (B) Their reciprocals are in A.P.
(C) Both (A) and (B) are true (D) Both (A) and (B) are false
2. If a, b, c be respectively pth, qth and rth terms of an A.P. the value of
a(q−r)+b(r− p)+c(p−q)is __________.
(A) 0 (B) 1 (C) –1 (D) None
3. If the pth term of an A.P. is q and the qth term is p the value of the rth term is_________.
(A) p – q – r (B) p + q – r
(C) p + q + r (D) None
4. If the pth term of an A.P. is q and the qth term is p the value of the (p + q)th term is_______.
(A) 0 (B) 1 (C) –1 (D) None
5. The sum of first n natural number is _______.
(A) (n/2)(n+1) (B) (n/6)(n+1)(2n+1)
(C) [(n/2)(n+1)]² (D) None
6. The sum of square of first n natural number is __________.
(A) (n/2)(n+1) (B) (n/6)(n+1)(2n+1)
(C) [(n/2)(n+1)]² (D) None
7. The sum of cubes of first n natural number is __________.
(A) (n/2)(n+1) (B) (n/6)(n+1)(2n+1)
(C) [(n/2)(n+1)]² (D) None
8. The sum of a series in A.P. is 72 the first term being 17 and the common difference –2. the
number of terms is __________.
(A) 6 (B) 12 (C) 6 or 12 (D) None
(cid:1)(cid:2)(cid:5)(cid:2)(cid:6)(cid:7)(cid:4)(cid:8)(cid:9)(cid:8)(cid:1)(cid:2)(cid:5)(cid:10)(cid:6)(cid:7)(cid:4)(cid:8)(cid:9)(cid:8)(cid:1)(cid:2)(cid:5)(cid:3)(cid:6)(cid:7)(cid:4)(cid:8)(cid:9)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)
9. Find the sum to n terms of
(A) ½(n–1) (B) ½(n+1) (C) (n–1) (D) (n+1)
10. If Sn the sum of first n terms in a series is given by 2n2 + 3n the series is in ______.
(A) A.P. (B) G.P. (C) H.P. (D) None
(cid:6)(cid:17)(cid:15)(cid:18)(cid:16) (cid:1)(cid:2)(cid:3)(cid:25)
Copyright -The Institute of Chartered Accountants of India
(cid:1)(cid:2)(cid:3)(cid:4)(cid:2)(cid:5)(cid:6)(cid:2)(cid:7)(cid:8)(cid:5)(cid:9)(cid:7)(cid:1)(cid:2)(cid:10)(cid:11)(cid:2)(cid:1)(cid:12)(cid:8)(cid:10)(cid:11)(cid:13)(cid:14)(cid:15)(cid:2)(cid:13)(cid:11)(cid:6)(cid:7)(cid:8)(cid:5)(cid:9)(cid:7)(cid:16)(cid:2)(cid:17)(cid:15)(cid:2)(cid:13)(cid:10)(cid:11)(cid:6)(cid:7)(cid:18)(cid:10)(cid:17)(cid:16)(cid:10)(cid:2)(cid:1)(cid:1)(cid:11)(cid:17)(cid:5)(cid:1)
11. The sum of all natural numbers between 200 and 400 which are divisible by 7 is ______.
(A) 7730 (B) 8729 (C) 7729 (D) 8730
12. The sum of natural numbers upto 200 excluding those divisible by 5 is ________.
(A) 20100 (B) 4100 (C) 16000 (D) None
13. If a, b, c be the sums of p q r terms respectively of an A.P. the value of
(cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:1)(cid:6)(cid:7)(cid:8)(cid:5)(cid:9)(cid:10)(cid:9)(cid:1)(cid:11)(cid:3)(cid:6)(cid:5)(cid:1)(cid:8)(cid:7)(cid:4)(cid:5)(cid:9)(cid:10)(cid:9)(cid:1)(cid:12)(cid:3)(cid:8)(cid:5)(cid:1)(cid:4)(cid:7)(cid:6)(cid:5)
is ______.
(A) 0 (B) 1 (C) –1 (D) None
(cid:13) (cid:12)(cid:13) (cid:12)(cid:13) (cid:14)(cid:12) (cid:10)(cid:14)(cid:12) (cid:3)(cid:14)
14. If be the respectively the sum of terms of an A.P. the value of
(cid:2) (cid:10) (cid:3)
(cid:13) (cid:13)(cid:1)(cid:13) (cid:5)(cid:13) (cid:4)
is given by ______.
(cid:3) (cid:10) (cid:2)
(A) 1 (B) 2 (C) 3 (D) None
15. The sum of n terms of two A.P.s are in the ratio of(cid:1)(cid:14)(cid:7)(cid:5)(cid:15)(cid:4)(cid:6)(cid:1)(cid:15)(cid:7)(cid:9)(cid:2)(cid:14)(cid:4). Then the _______ term of
the two series are equal.
(A) 12 (B) 6 (C) 3 (D) None
16. Find three numbers in A.P. whose sum is 6 and the product is –24
(A) –2, 2, 6 (B) –1, 1, 3 (C) 1, 3, 5 (D) 1, 4, 7
17. Find three numbers in A.P. whose sum is 6 and the sum of whose square is 44.
(A) –2, 2, 6 (B) –1, 1, 3 (C) 1, 3, 5 (D) 1, 4, 7
18. Find three numbers in A.P. whose sum is 6 and the sum of their cubes is 232.
(A) –2, 2, 6 (B) –1, 1, 3 (C) 1, 3, 5 (D) 1, 4, 7
19. Divide 12.50 into five parts in A.P. such that the first part and the last part are in the ratio
of 2:3
(A) 2, 2.25, 2.5, 2.75, 3 (B) –2, –2.25, –2.5, –2.75, –3
(C) 4, 4.5, 5, 5.5, 6 (D) –4, –4.5, –5, –5.5, –6
20. If a, b, c are in A.P. then the value of (cid:1)(cid:16)(cid:3)(cid:9)(cid:8)(cid:17)(cid:18)(cid:3)(cid:9)(cid:8)(cid:19)(cid:3)(cid:4)(cid:6)(cid:20)(cid:18)(cid:1)(cid:16)(cid:10)(cid:8)(cid:9)(cid:8)(cid:19)(cid:10)(cid:4)(cid:21) is
(A) 1 (B) 2 (C) 3 (D) None
21. If a, b, c are in A.P. then the value of (cid:1)(cid:16)(cid:10)(cid:9)(cid:8)(cid:17)(cid:16)(cid:19)(cid:8)(cid:9)(cid:8)(cid:19)(cid:10)(cid:4)(cid:6)(cid:1)(cid:16)(cid:18)(cid:8)(cid:9)(cid:8)(cid:18)(cid:19)(cid:8)(cid:9)(cid:8)(cid:19)(cid:16)(cid:4) is
(A) 1 (B) 2 (C) 3 (D) None
(cid:1)(cid:2)(cid:3)(cid:3) (cid:4)(cid:5)(cid:6)(cid:6)(cid:5)(cid:7)(cid:8) (cid:9)(cid:10)(cid:5)(cid:11)(cid:12)(cid:4)(cid:12)(cid:13)(cid:7)(cid:4)(cid:14)(cid:8) (cid:15)(cid:13)(cid:16)(cid:15)
Copyright -The Institute of Chartered Accountants of India
(cid:1)(cid:16)(cid:6)(cid:18)(cid:19)(cid:4)(cid:1)(cid:18)(cid:8)(cid:9)(cid:8)(cid:19)(cid:4)(cid:12)(cid:1)(cid:18)(cid:6)(cid:19)(cid:16)(cid:4)(cid:1)(cid:19)(cid:8)(cid:9)(cid:8)(cid:16)(cid:4)(cid:12)(cid:1)(cid:19)(cid:6)(cid:16)(cid:18)(cid:4)(cid:1)(cid:16)(cid:8)(cid:9)(cid:8)(cid:18)(cid:4)
22. If a, b, c are in A.P. then are in ____________.
(A) A.P. (B) G.P. (C) H.P. (D) None
23. If a, b, c are in A.P. then
(cid:16)(cid:10)(cid:1)(cid:18)(cid:8)(cid:9)(cid:8)(cid:19)(cid:4)(cid:12) (cid:18)(cid:10)(cid:1)(cid:19)(cid:8)(cid:9)(cid:8)(cid:16)(cid:4)(cid:12)(cid:19)(cid:10)(cid:1)(cid:16)(cid:8)(cid:9)(cid:8)(cid:18)(cid:4)
are in ________.
(A) A.P. (B) G.P. (C) H.P. (D) None
24. If
(cid:1)(cid:18)(cid:8)(cid:9)(cid:8)(cid:19)(cid:4)(cid:5)(cid:2)(cid:12)(cid:1)(cid:19)(cid:8)(cid:9)(cid:8)(cid:16)(cid:4)(cid:5)(cid:2)(cid:12)(cid:1)(cid:16)(cid:8)(cid:9)(cid:8)(cid:18)(cid:4)(cid:5)(cid:2)
are in A.P. then
(cid:16)(cid:10)(cid:12) (cid:18)(cid:10)(cid:12) (cid:19)(cid:10)
are in _________.
(A) A.P. (B) G.P. (C) H.P. (D) None
(cid:16)(cid:10)(cid:12) (cid:18)(cid:10)(cid:12) (cid:19)(cid:10) (cid:1)(cid:18)(cid:8)(cid:9)(cid:8)(cid:19)(cid:4)(cid:12)(cid:1)(cid:19)(cid:8)(cid:9)(cid:8)(cid:16)(cid:4)(cid:12)(cid:1)(cid:16)(cid:8)(cid:9)(cid:8)(cid:18)(cid:4)
25. If are in A.P. then are in ________.
(A) A.P. (B) G.P. (C) H.P. (D) None
(cid:16)(cid:10)(cid:12) (cid:18)(cid:10)(cid:12) (cid:19)(cid:10) (cid:16)(cid:6)(cid:1)(cid:18)(cid:8)(cid:9)(cid:8)(cid:19)(cid:4)(cid:12) (cid:18)(cid:6)(cid:1)(cid:19)(cid:8)(cid:9)(cid:8)(cid:16)(cid:4)(cid:12)(cid:19)(cid:6)(cid:1)(cid:16)(cid:8)(cid:9)(cid:8)(cid:18)(cid:4)
26. If are in A.P. then are in ____________.
(A) A.P. (B) G.P. (C) H.P. (D) None
27. If (cid:1)(cid:18)(cid:8)(cid:9)(cid:8)(cid:19)(cid:8)(cid:22)(cid:8)(cid:16)(cid:4)(cid:6)(cid:16)(cid:12)(cid:1)(cid:19)(cid:8)(cid:9)(cid:8)(cid:16)(cid:8)(cid:22)(cid:8)(cid:18)(cid:4)(cid:6)(cid:18)(cid:12)(cid:1)(cid:16)(cid:8)(cid:9)(cid:8)(cid:18)(cid:8)(cid:22)(cid:8)(cid:19)(cid:4)(cid:6)(cid:19) are in A.P. then (cid:16)(cid:12) (cid:18)(cid:12) (cid:19) are in __________.
(A) A.P. (B) G.P. (C) H.P. (D) None
28. If (cid:1)(cid:18)(cid:8)(cid:22)(cid:8)(cid:19)(cid:4)(cid:10)(cid:12)(cid:1)(cid:19)(cid:8)(cid:22)(cid:8)(cid:16)(cid:4)(cid:10)(cid:12)(cid:1)(cid:16)(cid:8)(cid:22)(cid:8)(cid:18)(cid:4)(cid:10) are in A.P. then (cid:1)(cid:18)(cid:8)(cid:22)(cid:8)(cid:19)(cid:4)(cid:12)(cid:1)(cid:19)(cid:8)(cid:22)(cid:8)(cid:16)(cid:4)(cid:12)(cid:1)(cid:16)(cid:8)(cid:22)(cid:8)(cid:18)(cid:4) are in _______.
(A) A.P. (B) G.P. (C) H.P. (D) None
29. If a b c are in A.P. then (cid:1)(cid:18)(cid:8)(cid:9)(cid:8)(cid:19)(cid:4)(cid:12)(cid:1)(cid:19)(cid:8)(cid:9)(cid:8)(cid:16)(cid:4)(cid:12)(cid:1)(cid:16)(cid:8)(cid:9)(cid:8)(cid:18)(cid:4) are in ________.
(A) A.P. (B) G.P. (C) H.P. (D) None
30. Find the number which should be added to the sum of any number of terms of the A.P.
3, 5, 7, 9, 11 …….resulting in a perfect square.
(A) –1 (B) 0 (C) 1 (D) None
31. The sum of n terms of an A.P. is (cid:10)(cid:7)(cid:10)(cid:9)(cid:8)(cid:3)(cid:7). Find the nth term.
(A) 4n + 1 (B) 4n - 1 (C) 2n + 1 (D) 2n - 1
32. The pth term of an A.P. is 1/q and the qth term is 1/p. The sum of the pqth term is_______.
(cid:2) (cid:2)
(cid:1)(cid:23)(cid:24)(cid:9)(cid:2)(cid:4) (cid:1)(cid:23)(cid:24)(cid:5)(cid:2)(cid:4) (cid:23)(cid:24)(cid:9)(cid:2) (cid:23)(cid:24)(cid:5)(cid:2)
(A) (B) (C) (D)
(cid:10) (cid:10)
(cid:6)(cid:17)(cid:15)(cid:18)(cid:16) (cid:1)(cid:2)(cid:3)(cid:19)
Copyright -The Institute of Chartered Accountants of India
(cid:1)(cid:2)(cid:3)(cid:4)(cid:2)(cid:5)(cid:6)(cid:2)(cid:7)(cid:8)(cid:5)(cid:9)(cid:7)(cid:1)(cid:2)(cid:10)(cid:11)(cid:2)(cid:1)(cid:12)(cid:8)(cid:10)(cid:11)(cid:13)(cid:14)(cid:15)(cid:2)(cid:13)(cid:11)(cid:6)(cid:7)(cid:8)(cid:5)(cid:9)(cid:7)(cid:16)(cid:2)(cid:17)(cid:15)(cid:2)(cid:13)(cid:10)(cid:11)(cid:6)(cid:7)(cid:18)(cid:10)(cid:17)(cid:16)(cid:10)(cid:2)(cid:1)(cid:1)(cid:11)(cid:17)(cid:5)(cid:1)
33. The sum of p terms of an A.P. is q and the sum of q terms is p. The sum of p + q terms is
________.
(A) – (p + q) (B) p + q (C) (p – q)2 (D) P2 – q2
34. If S S S be the sums of n terms of three A.P.s the first term of each being unity and the
1, 2, 3
respective common differences 1, 2, 3 then (S + S ) / S is ______.
1 3 2
(A) 1 (B) 2 (C) –1 (D) None
35. The sum of all natural numbers between 500 and 1000, which are divisible by 13, is _______.
(A) 28400 (B) 28405 (C) 28410 (D) None
36. The sum of all natural numbers between 100 and 300, which are divisible by 4, is ____.
(A) 10200 (B) 30000 (C) 8200 (D) 2200
37. The sum of all natural numbers from 100 to 300 excluding those, which are divisible by 4, is
_______.
(A) 10200 (B) 30000 (C) 8200 (D) 2200
38. The sum of all natural numbers from 100 to 300, which are divisible by 5, is ______.
(A) 10200 (B) 30000 (C) 8200 (D) 2200
39. The sum of all natural numbers from 100 to 300, which are divisible by 4 and 5, is ______.
(A) 10200 (B) 30000 (C) 8200 (D) 2200
40. The sum of all natural numbers from 100 to 300, which are divisible by 4 or 5, is ______.
(A) 10200 (B) 8200 (C) 2200 (D) 16200
41. If the n terms of two A.P.s are in the ratio (3n+4) : (n+4) the ratio of the fourth term
is ______.
(A) 2 (B) 3 (C) 4 (D) None
42. If a, b, c, d are in A.P. then
(A) (cid:16)(cid:10)(cid:22)(cid:8)(cid:3)(cid:18)(cid:10)(cid:9)(cid:3)(cid:19)(cid:10)(cid:22)(cid:8)(cid:25)(cid:10)(cid:26)(cid:27) (B) (cid:16)(cid:10)(cid:9)(cid:3)(cid:18)(cid:10)(cid:9)(cid:3)(cid:19)(cid:10)(cid:9)(cid:25)(cid:10)(cid:26)(cid:27) (C) (cid:16)(cid:10)(cid:9)(cid:8)(cid:3)(cid:18)(cid:10)(cid:9)(cid:8)(cid:3)(cid:19)(cid:10)(cid:22)(cid:8)(cid:25)(cid:10)(cid:26)(cid:27) (D) None
43. If a, b, c, d, e are in A.P. then
(A) a – b – d + e = 0 (B) a – 2c + e = 0 (C) b – 2c + d = 0 (D) all the above
44. The three numbers in A.P. whose sum is 18 and product is 192 are _______.
(A) 4, 6, 8 (B) –4, –6, –8 (C) 8, 6, 4
(D) both (A) and (C)
(cid:1)(cid:2)(cid:3)(cid:20) (cid:4)(cid:5)(cid:6)(cid:6)(cid:5)(cid:7)(cid:8) (cid:9)(cid:10)(cid:5)(cid:11)(cid:12)(cid:4)(cid:12)(cid:13)(cid:7)(cid:4)(cid:14)(cid:8) (cid:15)(cid:13)(cid:16)(cid:15)
Copyright -The Institute of Chartered Accountants of India
45. The three numbers in A.P., whose sum is 27 and the sum of their squares is 341, are _______.
(A) 2, 9, 16 (B) 16, 9, 2 (C) both (A) and (B) (D) –2, –9, –16
46. The four numbers in A.P., whose sum is 24 and their product is 945, are _______.
(A) 3, 5, 7, 9 (B) 2, 4, 6, 8 (C) 5, 9, 13, 17 (D) None
47. The four numbers in A.P., whose sum is 20 and the sum of their squares is 120, are _______.
(A) 3, 5, 7, 9 (B) 2, 4, 6, 8 (C) 5, 9, 13, 17 (D) None
48. The four numbers in A.P. with the sum of second and third being 22 and the product of
the first and fourth beinf 85 are _______.
(A) 3, 5, 7, 9 (B) 2, 4, 6, 8 (C) 5, 9, 13, 17 (D) None
49. The five numbers in A.P. with their sum 25 and the sum of their squares 135 are _______.
(A) 3, 4, 5, 6, 7 (B) 3, 3.5, 4, 4.5, 5 (C) –3, –4, –5, –6, –7
(D) –3, –3.5, –4, –4.5, –5
50. The five numbers in A.P. with the sum 20 and product of the first and last 15 are _______.
(A) 3, 4, 5, 6, 7 (B) 3, 3.5, 4, 4.5, 5 (C) –3, –4, –5, –6, –7
(D) –3, –3.5, –4, –4.5, –5
51. The sum of n terms of 2, 4, 6, 8….. is
(A) n(n+1) (B) (n/2)(n+1) (C) n(n–1) (D) (n/2)(n-1)
52. The sum of n terms of a+b, 2a, 3a–b, ….. is
(A) n(a–b)+2b (B) n(a+b) (C) both the above (D) None
53. The sum of n terms of (x + y)², (x² + y²), (x – y)²,...... is
(A) (x + y)² –2(n – 1)xy (B) n(x + y)² – n(n – 1)xy(C) both the above (D) None
54. The sum of n terms of (1/n)(n–1), (1/n) (n–2), (1/n) (n–3)....... is
(A) 0 (B) (1/2)(n–1) (C) (1/2)(n+1) (D) None
55. The sum of n terms of 1.4, 3.7, 5.10 ……. Is
(A) (n/2)(4n²+5n–1) (B) n(4n²+5n–1) (C) (n/2)(4n²–5n–1)(D) None
(cid:6)(cid:17)(cid:15)(cid:18)(cid:16) (cid:1)(cid:2)(cid:3)(cid:21)
Copyright -The Institute of Chartered Accountants of India
(cid:1)(cid:2)(cid:3)(cid:4)(cid:2)(cid:5)(cid:6)(cid:2)(cid:7)(cid:8)(cid:5)(cid:9)(cid:7)(cid:1)(cid:2)(cid:10)(cid:11)(cid:2)(cid:1)(cid:12)(cid:8)(cid:10)(cid:11)(cid:13)(cid:14)(cid:15)(cid:2)(cid:13)(cid:11)(cid:6)(cid:7)(cid:8)(cid:5)(cid:9)(cid:7)(cid:16)(cid:2)(cid:17)(cid:15)(cid:2)(cid:13)(cid:10)(cid:11)(cid:6)(cid:7)(cid:18)(cid:10)(cid:17)(cid:16)(cid:10)(cid:2)(cid:1)(cid:1)(cid:11)(cid:17)(cid:5)(cid:1)
56. The sum of n terms of 1², 3², 5², 7²,.......is
(A) (n/3)(4n² –1) (B) (n/2)(4n² –1) (C) (n/3)(4n² +1) (D) None
57. The sum of n terms of 1, (1 + 2), (1 + 2 + 3) …….. is
(A) (n/3)(n+1)(n–2) (B) (n/3)(n+1)(n+2) (C) n(n+1)(n+2) (D) None
58. The sum of n terms of the series 1²/1+(1²+2²)/2+(1²+2²+3²)/3+.......is
(A) (n/36)(4n² +15n+17) (B) (n/12)(4n²+15n+17)
(C) (n/12)(4n² +15n+17) (D) None
59. The sum of n terms of the series 2.4.6 + 4.6.8 + 6.8.10 + ………. is
(A) 2n(n³+6n²+11n+6) (B) 2n(n³–6n²+11n–6)
(C) n(n³+6n²+11n+6) (D) n(n³+6n²+11n–6)
60. The sum of n terms of the series (cid:2)(cid:11)(cid:3)(cid:10)(cid:9)(cid:17)(cid:11)(cid:17)(cid:10)(cid:9)(cid:14)(cid:11)(cid:15)(cid:10)(cid:9)(cid:2)(cid:27)(cid:11)(cid:28)(cid:10)(cid:9)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)is
(A) (n/12)(n+1)(9n²+49n+44)–8n (B) (n/12)(n+1)(9n²+49n+44)+8n
(C) (n/6)(2n+1)(9n²+49n+44)–8n (D) None
61. The sum of n terms of the series 4 + 6 + 9 + 13 …….. is
(A) (n/6)(n²+3n+20) (B) (n/6)(n+1)(n+2) (C) (n/3)(n+1)(n+2) (D) None
62. The sum to n terms of the series 11, 23, 59, 167 ………is
(A) 3n+1+5n–3 (B) 3n+1+5n+3 (C) 3n+5n–3 (D) None
(cid:2)(cid:6)(cid:1)(cid:17)(cid:11)(cid:29)(cid:4)(cid:9)(cid:2)(cid:6)(cid:1)(cid:29)(cid:11)(cid:2)(cid:17)(cid:4)(cid:9)(cid:2)(cid:6)(cid:1)(cid:2)(cid:17)(cid:11)(cid:2)(cid:29)(cid:4)(cid:9)(cid:2)(cid:6)(cid:1)(cid:2)(cid:29)(cid:11)(cid:10)(cid:17)(cid:4)(cid:9)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)
63. The sum of n terms of the series is
(A) (n/4)(5n+4)–1 (B) (n/4)(5n+4) (C) (n/4)(5n–4)–1 (D) None
64. The sum of n terms of the series 1 + 3 + 5 + ………. Is
(A) (cid:14)(cid:10) (B) (cid:10)(cid:14)(cid:10) (C) (cid:14)(cid:10)(cid:6)(cid:10) (D) None
65. The sum of n terms of the series 2 + 6 + 10 + ……. is
(A) (cid:10)(cid:14)(cid:10) (B) (cid:14)(cid:10) (C) (cid:14)(cid:10)(cid:6)(cid:10) (D) (cid:17)(cid:14)(cid:10)
66. The sum of n terms of the series 1.2 + 2.3 + 3.4 + ……. Is
(A) (n/3)(n+1)(n+2) (B) (n/2)(n+1)(n+2) (C) (n/3)(n+1)(n– 2)(D) None
(cid:1)(cid:2)(cid:3)(cid:1) (cid:4)(cid:5)(cid:6)(cid:6)(cid:5)(cid:7)(cid:8) (cid:9)(cid:10)(cid:5)(cid:11)(cid:12)(cid:4)(cid:12)(cid:13)(cid:7)(cid:4)(cid:14)(cid:8) (cid:15)(cid:13)(cid:16)(cid:15)
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67. The sum of n terms of the series 1.2.3 + 2.3.4 + 3.4.5 + …….is
(A) (n/4)(n+1)(n+2)(n+3) (B) (n/3)(n+1)(n+2)(n+3)
(C) (n/2)(n+1)(n+2)(n+3) (D) None
68. The sum of n terms of the series 1.2+3.2²+5.2³+7.24+....... is
(A) (n–1)2n+2–2n+1 +6 (B) (n+1)2n+2–2n+1 +6 (C) (n–1)2n+2–2n+1 –6 (D) None
69. The sum of n terms of the series 1/(3.8)+1/(8.13)+1/(13.18)+...... is
(A) (n/3)(5n+3)–1 (B) (n/2)(5n+3)–1 (C) (n/2)(5n–3)–1 (D) None
70. The sum of n terms of the series 1/1+1/(1+2)+1/(1+2+3)+..... is
(A) 2n(n+1)–1 (B) n(n+1) (C) 2n(n–1)–1 (D) None
71. The sum of n terms of the series 22+52+82+........ is
(A) (n/2)(6n2+3n–1) (B) (n/2)(6n2–3n–1)
(C) (n/2)(6n2+3n+1) (D) None
72. The sum of n terms of the series 12+32+52+........ is
(cid:14)
(cid:8)(cid:1)(cid:17)(cid:14)(cid:10)(cid:22)(cid:8)(cid:2)(cid:4)
(A) (cid:3) (B) n2(2n2+1) (C) n(2n–1) (D) n(2n+1)
73. The sum of n terms of the series 1.4 + 3.7 + 5.10 + …… is
(A) (n/2)(4n2+5 1) (B) (n/2)(5n2+4n–1)
(C) (n/2)(4n2+5n+1) (D) None
(cid:10)(cid:11)(cid:3)(cid:10)(cid:9)(cid:15)(cid:11)(cid:17)(cid:10)(cid:9)(cid:30)(cid:11)(cid:15)(cid:10)(cid:9)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)
74. The sum of n terms of the series is
(A)
(cid:1)(cid:14)(cid:6)(cid:2)(cid:10)(cid:4)(cid:1)(cid:29)(cid:14)(cid:3)(cid:9)(cid:28)(cid:10)(cid:14)(cid:10)(cid:9)(cid:2)(cid:10)(cid:3)(cid:14)(cid:9)(cid:10)(cid:10)(cid:4)
(B)
(cid:1)(cid:14)(cid:6)(cid:2)(cid:10)(cid:4)(cid:1)(cid:29)(cid:14)(cid:3)(cid:5)(cid:28)(cid:10)(cid:14)(cid:10)(cid:9)(cid:2)(cid:10)(cid:3)(cid:14)(cid:5)(cid:10)(cid:10)(cid:4)
(C)
(cid:1)(cid:14)(cid:6)(cid:28)(cid:4)(cid:1)(cid:29)(cid:14)(cid:3)(cid:9)(cid:28)(cid:10)(cid:14)(cid:10)(cid:9)(cid:2)(cid:10)(cid:3)(cid:14)(cid:9)(cid:10)(cid:10)(cid:4)
(D) None
75. The sum of n terms of the series 1 + (1 + 3) + (1 + 3 + 5) + ……. is
(cid:1)(cid:14)(cid:6)(cid:28)(cid:4)(cid:1)(cid:14)(cid:9)(cid:2)(cid:4)(cid:1)(cid:10)(cid:14)(cid:9)(cid:2)(cid:4) (cid:1)(cid:14)(cid:6)(cid:28)(cid:4)(cid:1)(cid:14)(cid:9)(cid:2)(cid:4)(cid:1)(cid:14)(cid:9)(cid:10)(cid:4) (cid:1)(cid:14)(cid:6)(cid:3)(cid:4)(cid:1)(cid:14)(cid:9)(cid:2)(cid:4)(cid:1)(cid:10)(cid:14)(cid:9)(cid:2)(cid:4)
(A) (B) (C) (D) None
76. The sum of n terms of the series
(cid:2)(cid:10)(cid:9)(cid:1)(cid:2)(cid:10)(cid:9)(cid:10)(cid:10)(cid:4)(cid:9)(cid:1)(cid:2)(cid:10)(cid:9)(cid:10)(cid:10)(cid:9)(cid:3)(cid:10)(cid:4)(cid:9)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)
is
(A)
(cid:1)(cid:14)(cid:6)(cid:2)(cid:10)(cid:4)(cid:1)(cid:14)(cid:9)(cid:2)(cid:4)(cid:10)(cid:1)(cid:14)(cid:9)(cid:10)(cid:4)
(B)
(cid:1)(cid:14)(cid:6)(cid:2)(cid:10)(cid:4)(cid:1)(cid:14)(cid:5)(cid:2)(cid:4)(cid:10)(cid:1)(cid:14)(cid:9)(cid:10)(cid:4)
(C)
(cid:1)(cid:14)(cid:6)(cid:2)(cid:10)(cid:4)(cid:1)(cid:14)(cid:10)(cid:5)(cid:2)(cid:4)(cid:1)(cid:14)(cid:9)(cid:10)(cid:4)
(D) None
(cid:6)(cid:17)(cid:15)(cid:18)(cid:16) (cid:1)(cid:2)(cid:3)(cid:22)
Copyright -The Institute of Chartered Accountants of India
(cid:1)(cid:2)(cid:3)(cid:4)(cid:2)(cid:5)(cid:6)(cid:2)(cid:7)(cid:8)(cid:5)(cid:9)(cid:7)(cid:1)(cid:2)(cid:10)(cid:11)(cid:2)(cid:1)(cid:12)(cid:8)(cid:10)(cid:11)(cid:13)(cid:14)(cid:15)(cid:2)(cid:13)(cid:11)(cid:6)(cid:7)(cid:8)(cid:5)(cid:9)(cid:7)(cid:16)(cid:2)(cid:17)(cid:15)(cid:2)(cid:13)(cid:10)(cid:11)(cid:6)(cid:7)(cid:18)(cid:10)(cid:17)(cid:16)(cid:10)(cid:2)(cid:1)(cid:1)(cid:11)(cid:17)(cid:5)(cid:1)
77. The sum of n terms of the series
(cid:2)(cid:9)(cid:1)(cid:2)(cid:9)(cid:2)(cid:6)(cid:3)(cid:4)(cid:9)(cid:1)(cid:2)(cid:9)(cid:2)(cid:6)(cid:3)(cid:9)(cid:2)(cid:6)(cid:3)(cid:10)(cid:4)(cid:9)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)is
(A)
(cid:1)(cid:3)(cid:6)(cid:10)(cid:4)(cid:1)(cid:2)(cid:5)(cid:3)(cid:5)(cid:7)(cid:4)
(B)
(cid:1)(cid:3)(cid:6)(cid:10)(cid:4)(cid:20)(cid:14)(cid:5)(cid:1)(cid:2)(cid:6)(cid:10)(cid:4)(cid:1)(cid:2)(cid:5)(cid:3)(cid:5)(cid:7)(cid:4)(cid:21)
(C) Both (D) None
(cid:14)(cid:11)(cid:2)(cid:9)(cid:1)(cid:14)(cid:5)(cid:2)(cid:4)(cid:11)(cid:10)(cid:9)(cid:1)(cid:14)(cid:5)(cid:10)(cid:4)(cid:11)(cid:3)(cid:9) (cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)
78. The sum of n terms of the series is
(cid:1)(cid:14)(cid:6)(cid:28)(cid:4)(cid:1)(cid:14)(cid:9)(cid:2)(cid:4)(cid:1)(cid:14)(cid:9)(cid:10)(cid:4) (cid:1)(cid:14)(cid:6)(cid:3)(cid:4)(cid:1)(cid:14)(cid:9)(cid:2)(cid:4)(cid:1)(cid:14)(cid:9)(cid:10)(cid:4) (cid:1)(cid:14)(cid:6)(cid:10)(cid:4)(cid:1)(cid:14)(cid:9)(cid:2)(cid:4)(cid:1)(cid:14)(cid:9)(cid:10)(cid:4)
(A) (B) (C) (D) None
79. The sum of n terms of the series 1 + 5 + 12 + 22 + ….. is
(A)
(cid:1)(cid:14)(cid:10)(cid:6)(cid:10)(cid:4)(cid:1)(cid:14)(cid:9)(cid:2)(cid:4)
(B)
(cid:14)(cid:10)(cid:1)(cid:14)(cid:9)(cid:2)(cid:4)
(C)
(cid:1)(cid:14)(cid:10)(cid:6)(cid:10)(cid:4)(cid:1)(cid:14)(cid:5)(cid:2)(cid:4)
(D) None
80. The sum of n terms of the series 4 + 14 + 30 + 52 + 80 + …… is
(A)
(cid:14)(cid:1)(cid:14)(cid:9)(cid:2)(cid:4)(cid:10)
(B)
(cid:14)(cid:1)(cid:14)(cid:5)(cid:2)(cid:4)(cid:10)
(C)
(cid:14)(cid:1)(cid:14)(cid:10)(cid:5)(cid:2)(cid:4)
(D) None
81. The sum of n terms of the series 3 + 6 + 11 + 20 + 37 + …….. is
(A)
(cid:10)(cid:14)(cid:9)(cid:2)(cid:9)(cid:1)(cid:14)(cid:6)(cid:10)(cid:4)(cid:1)(cid:14)(cid:9)(cid:2)(cid:4)(cid:5)(cid:10)
(B)
(cid:10)(cid:14)(cid:9)(cid:2)(cid:9)(cid:1)(cid:14)(cid:6)(cid:10)(cid:4)(cid:1)(cid:14)(cid:9)(cid:2)(cid:4)(cid:5)(cid:2)
(C)
(cid:10)(cid:14)(cid:9)(cid:2)(cid:9)(cid:1)(cid:14)(cid:6)(cid:10)(cid:4)(cid:1)(cid:14)(cid:5)(cid:2)(cid:4)(cid:5)(cid:10)
(D) None
82. The nth terms of the series is 1/(4.7) + 1/(7.10) + 1/(10.13) + ……. is
(A)
(cid:1)(cid:2)(cid:6)(cid:3)(cid:4)(cid:20)(cid:1)(cid:3)(cid:14)(cid:9)(cid:2)(cid:4)(cid:5)(cid:2)(cid:5)(cid:1)(cid:3)(cid:14)(cid:9)(cid:17)(cid:4)(cid:5)(cid:2)(cid:21)
(B)
(cid:1)(cid:2)(cid:6)(cid:3)(cid:4)(cid:20)(cid:1)(cid:3)(cid:14)(cid:5)(cid:2)(cid:4)(cid:5)(cid:2)(cid:5)(cid:1)(cid:3)(cid:14)(cid:9)(cid:17)(cid:4)(cid:5)(cid:2)(cid:21)
(C)
(cid:1)(cid:2)(cid:6)(cid:3)(cid:4)(cid:20)(cid:1)(cid:3)(cid:14)(cid:9)(cid:2)(cid:4)(cid:5)(cid:2)(cid:5)(cid:1)(cid:3)(cid:14)(cid:5)(cid:17)(cid:4)(cid:5)(cid:2)(cid:21)
(D) None
83. In question No.(82) the sum of the series upto µ is
(A)
(cid:1)(cid:14)(cid:6)(cid:17)(cid:4)(cid:1)(cid:3)(cid:14)(cid:9)(cid:17)(cid:4)(cid:5)(cid:2)
(B)
(cid:1)(cid:14)(cid:6)(cid:17)(cid:4)(cid:1)(cid:3)(cid:14)(cid:5)(cid:17)(cid:4)(cid:5)(cid:2)
(C)
(cid:1)(cid:14)(cid:6)(cid:10)(cid:4)(cid:1)(cid:3)(cid:14)(cid:9)(cid:17)(cid:4)(cid:5)(cid:2)
(D) None
84. The sum of n terms of the series
(cid:2)(cid:10)(cid:6)(cid:2)(cid:9)(cid:1)(cid:2)(cid:10)(cid:9)(cid:10)(cid:10)(cid:4)(cid:6)(cid:1)(cid:2)(cid:9)(cid:10)(cid:4)(cid:9)(cid:1)(cid:2)(cid:10)(cid:9)(cid:10)(cid:10)(cid:9)(cid:3)(cid:10)(cid:4)(cid:6)(cid:1)(cid:2)(cid:9)(cid:10)(cid:9)(cid:3)(cid:4)(cid:9)(cid:11)(cid:11)(cid:11)(cid:11)is
(cid:1)(cid:14)(cid:6)(cid:3)(cid:4)(cid:1)(cid:14)(cid:9)(cid:10)(cid:4) (cid:1)(cid:14)(cid:6)(cid:3)(cid:4)(cid:1)(cid:14)(cid:9)(cid:2)(cid:4) (cid:1)(cid:14)(cid:6)(cid:3)(cid:4)(cid:1)(cid:14)(cid:9)(cid:3)(cid:4)
(A) (B) (C) (D) None
85. The sum of n terms of the series
(cid:2)(cid:3)(cid:6)(cid:2)(cid:9)(cid:1)(cid:2)(cid:3)(cid:9)(cid:10)(cid:3)(cid:4)(cid:6)(cid:10)(cid:9)(cid:1)(cid:2)(cid:3)(cid:9)(cid:10)(cid:3)(cid:9)(cid:3)(cid:3)(cid:4)(cid:6)(cid:3)(cid:9)(cid:11)(cid:11)(cid:11)(cid:11)
is
(cid:1)(cid:14)(cid:6)(cid:17)(cid:30)(cid:4)(cid:1)(cid:14)(cid:9)(cid:2)(cid:4)(cid:1)(cid:14)(cid:9)(cid:10)(cid:4)(cid:1)(cid:3)(cid:14)(cid:9)(cid:15)(cid:4) (cid:1)(cid:14)(cid:6)(cid:10)(cid:17)(cid:4)(cid:1)(cid:14)(cid:9)(cid:2)(cid:4)(cid:1)(cid:14)(cid:9)(cid:10)(cid:4)(cid:1)(cid:3)(cid:14)(cid:9)(cid:15)(cid:4)
(A) (B)
(cid:1)(cid:7)(cid:6)(cid:17)(cid:30)(cid:4)(cid:1)(cid:7)(cid:9)(cid:2)(cid:4)(cid:1)(cid:7)(cid:9)(cid:10)(cid:4)(cid:1)(cid:15)(cid:7)(cid:9)(cid:3)(cid:4)
(C) (D) None
86. The value of
(cid:7)(cid:10)(cid:9)(cid:8)(cid:9)(cid:10)(cid:7)(cid:20)(cid:2)(cid:9)(cid:10)(cid:9)(cid:3)(cid:9)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:9)(cid:1)(cid:7)(cid:5)(cid:2)(cid:4)(cid:21)
is
(A) n3 (B) n2 (C) n (D) None
87. (cid:10)(cid:17)(cid:7)(cid:5)(cid:2) is divisible by
(A) 15 (B) 4 (C) 6 (D) 64
(cid:1)(cid:2)(cid:3)(cid:23) (cid:4)(cid:5)(cid:6)(cid:6)(cid:5)(cid:7)(cid:8) (cid:9)(cid:10)(cid:5)(cid:11)(cid:12)(cid:4)(cid:12)(cid:13)(cid:7)(cid:4)(cid:14)(cid:8) (cid:15)(cid:13)(cid:16)(cid:15)
Copyright -The Institute of Chartered Accountants of India
88. (cid:3)(cid:7)(cid:5)(cid:10)(cid:14)(cid:5)(cid:2) is divisible by
(A) 15 (B) 4 (C) 6 (D) 64
(cid:14)(cid:1)(cid:14)(cid:5)(cid:2)(cid:4)(cid:1)(cid:10)(cid:14)(cid:5)(cid:2)(cid:4)
89. is divisible by
(A) 15 (B) 4 (C) 6 (D) 64
90. (cid:14)(cid:10)(cid:7)(cid:9)(cid:2)(cid:28)(cid:14)(cid:5)(cid:2) is divisible by
(A) 15 (B) 4 (C) 6 (D) 64
91. The sum of n terms of the series whose nth term
(cid:3)(cid:7)(cid:10)(cid:9)(cid:10)(cid:7)
is is given by
(cid:1)(cid:14)(cid:6)(cid:10)(cid:4)(cid:1)(cid:14)(cid:9)(cid:2)(cid:4)(cid:1)(cid:10)(cid:14)(cid:9)(cid:3)(cid:4) (cid:1)(cid:14)(cid:6)(cid:10)(cid:4)(cid:1)(cid:14)(cid:9)(cid:2)(cid:4)(cid:1)(cid:3)(cid:14)(cid:9)(cid:10)(cid:4)
(A) (B)
(cid:1)(cid:14)(cid:6)(cid:10)(cid:4)(cid:1)(cid:14)(cid:9)(cid:2)(cid:4)(cid:1)(cid:3)(cid:14)(cid:5)(cid:10)(cid:4) (cid:1)(cid:7)(cid:6)(cid:10)(cid:4)(cid:1)(cid:7)(cid:9)(cid:2)(cid:4)(cid:1)(cid:10)(cid:7)(cid:5)(cid:3)(cid:4)
(C) (D)
92. The sum of n terms of the series whose nth term (cid:7)(cid:11)(cid:10)(cid:7)is is given by
(cid:1)(cid:7)(cid:5)(cid:2)(cid:4)(cid:10)(cid:7)(cid:9)(cid:2)(cid:9)(cid:10) (cid:1)(cid:7)(cid:9)(cid:2)(cid:4)(cid:10)(cid:7)(cid:9)(cid:2)(cid:9)(cid:10) (cid:1)(cid:7)(cid:5)(cid:2)(cid:4)(cid:10)(cid:7)(cid:9)(cid:10)
(A) (B) (C) (D) None
93. The sum of n terms of the series whose nth term (cid:15)(cid:11)(cid:3)(cid:7)(cid:9)(cid:2)(cid:9)(cid:10)(cid:7) is is given by
(cid:1)(cid:15)(cid:6)(cid:10)(cid:4)(cid:1)(cid:3)(cid:7)(cid:9)(cid:10)(cid:5)(cid:29)(cid:4)(cid:9)(cid:7)(cid:1)(cid:7)(cid:9)(cid:2)(cid:4) (cid:1)(cid:10)(cid:6)(cid:15)(cid:4)(cid:1)(cid:3)(cid:7)(cid:9)(cid:10)(cid:5)(cid:29)(cid:4)(cid:9)(cid:7)(cid:1)(cid:7)(cid:9)(cid:2)(cid:4)
(A) (B)
(cid:1)(cid:15)(cid:6)(cid:10)(cid:4)(cid:1)(cid:3)(cid:7)(cid:9)(cid:10)(cid:9)(cid:29)(cid:4)(cid:9)(cid:7)(cid:1)(cid:7)(cid:9)(cid:2)(cid:4)
(C) (D) None
94. If the third term of a G.P. is the square of the first and the fifth term is 64 the series would
be ________.
(A) 4 + 8 + 16 + 32 + …. (B) 4 – 8 + 16 – 32 + ……..
(C) both (D) None
95. Three numbers whose sum is 15 are in A.P. but if they are added by 1, 4, 19 respectively
they are in G.P. The numbers are _______.
(A) 2, 5, 8 (B) 26, 5, –16 (C) Both (D) None
(cid:16)(cid:24)(cid:5)(cid:31)(cid:11)(cid:18)(cid:31)(cid:5)(cid:23)(cid:11)(cid:19)(cid:23)(cid:5)(cid:24)
96. If a, b, c are the pth, qth and rth terms of a G.P. respectively the value of
is ________
(A) 0 (B) 1 (C) –1 (D) None
(cid:18)(cid:5)(cid:19)(cid:11)!(cid:19)(cid:5)(cid:16)(cid:11)"(cid:16)(cid:5)(cid:18)
97. If a, b, c are in A.P. and x, y, z in G.P. then the value of is ________
(A) 0 (B) 1 (C) –1 (D) None
(cid:6)(cid:17)(cid:15)(cid:18)(cid:16) (cid:1)(cid:2)(cid:3)(cid:24)
Copyright -The Institute of Chartered Accountants of India
(cid:1)(cid:2)(cid:3)(cid:4)(cid:2)(cid:5)(cid:6)(cid:2)(cid:7)(cid:8)(cid:5)(cid:9)(cid:7)(cid:1)(cid:2)(cid:10)(cid:11)(cid:2)(cid:1)(cid:12)(cid:8)(cid:10)(cid:11)(cid:13)(cid:14)(cid:15)(cid:2)(cid:13)(cid:11)(cid:6)(cid:7)(cid:8)(cid:5)(cid:9)(cid:7)(cid:16)(cid:2)(cid:17)(cid:15)(cid:2)(cid:13)(cid:10)(cid:11)(cid:6)(cid:7)(cid:18)(cid:10)(cid:17)(cid:16)(cid:10)(cid:2)(cid:1)(cid:1)(cid:11)(cid:17)(cid:5)(cid:1)
(cid:1) (cid:18)(cid:11)!(cid:19)(cid:11)"(cid:16)(cid:4)(cid:13)(cid:1) (cid:19)(cid:11)!(cid:16)(cid:11)"(cid:18)(cid:4)
98. If a, b, c are in A.P. and x, y, z in G.P. then the value of is ____
(A) 0 (B) 1 (C) –1 (D) None
99. The sum of n terms of the series 7 + 77 + 777 + …… is
(cid:1)(cid:14)(cid:6)(cid:29)(cid:4)(cid:20)(cid:1)(cid:2)(cid:6)(cid:29)(cid:4)(cid:1)(cid:2)(cid:27)(cid:7)(cid:9)(cid:2)(cid:5)(cid:2)(cid:27)(cid:4)(cid:5)(cid:7)(cid:21) (cid:1)(cid:29)(cid:6)(cid:2)(cid:27)(cid:4)(cid:20)(cid:1)(cid:2)(cid:6)(cid:29)(cid:4)(cid:1)(cid:2)(cid:27)(cid:7)(cid:9)(cid:2)(cid:5)(cid:2)(cid:27)(cid:4)(cid:5)(cid:7)(cid:21)
(A) (B)
(cid:1)(cid:2)(cid:27)(cid:6)(cid:29)(cid:4)(cid:20)(cid:1)(cid:2)(cid:6)(cid:29)(cid:4)(cid:1)(cid:2)(cid:27)(cid:7)(cid:9)(cid:2)(cid:5)(cid:2)(cid:27)(cid:4)(cid:5)(cid:7)(cid:21)
(C) (D) None
100.The least value of n for which the sum of n terms of the series 1 + 3 + 32+ …… is greater
than 7000 is ______.
(A) 9 (B) 10 (C) 8 (D) 7
101.If ‘S’ be the sum, ‘P’ the product and ‘R’ the sum of the reciprocals of n terms in a G.P.
then ‘P’ is the _______ of Sn and R-n.
(A) Arithmetic Mean (B) Geometric Mean (C) Harmonic Mean (D) None
(cid:30)(cid:9)(cid:17) (cid:10)(cid:9)(cid:17)(cid:9)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)
102.Sum upto of the series is
(A) (cid:30)(cid:1)(cid:10)(cid:9) (cid:10)(cid:4) (B) (cid:30)(cid:1)(cid:10)(cid:5) (cid:10)(cid:4) (C) (cid:17)(cid:1)(cid:10)(cid:9) (cid:10)(cid:4) (D) (cid:17)(cid:1)(cid:10)(cid:5) (cid:10)(cid:4)
(cid:2)(cid:6)(cid:10)(cid:9)(cid:2)(cid:6)(cid:3)(cid:10)(cid:9)(cid:2)(cid:6)(cid:10)(cid:3)(cid:9)(cid:2)(cid:6)(cid:3)(cid:17)(cid:9)(cid:2)(cid:6)(cid:10)(cid:15)(cid:9)(cid:2)(cid:6)(cid:3)(cid:28)(cid:9)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)
103.Sum upto of the series is
(A) 19/24 (B) 24/19 (C) 5/24 (D) None
104.If (cid:2)(cid:9)(cid:16)(cid:9)(cid:16)(cid:10)(cid:9)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11) a (cid:26) (cid:8) and (cid:2)(cid:9)(cid:18)(cid:9)(cid:18)(cid:10)(cid:9)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11) a (cid:26)! then (cid:2)(cid:9)(cid:16)(cid:18)(cid:9)(cid:16)(cid:10)(cid:18)(cid:10)(cid:9)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11) a is given by
__________.
(cid:1) !(cid:4)(cid:6)(cid:1) (cid:9)!(cid:5)(cid:2)(cid:4) (cid:1) !(cid:4)(cid:6)(cid:1) (cid:5)!(cid:5)(cid:2)(cid:4) (cid:1) !(cid:4)(cid:6)(cid:1) (cid:9)!(cid:9)(cid:2)(cid:4)
(A) (B) (C) (D) None
105. If the sum of three numbers in G.P. is 35 and their product is 1000 the numbers are ____.
(A) 20, 10, 5 (B) 5, 10, 20 (C) both (D) None
106.If the sum of three numbers in G.P. is 21 and the sum of their squares is 189 the numbers
are ____.
(A) 3, 6, 12 (B) 12, 6, 3 (C) both (D) None
(cid:16)(cid:1)(cid:18)(cid:10)(cid:9)(cid:19)(cid:10)(cid:4)(cid:5)(cid:19)(cid:1)(cid:16)(cid:10)(cid:9)(cid:18)(cid:10)(cid:4)
107.If a, b, c are in G.P. then the value of is ____
(A) 0 (B) 1 (C) –1 (D) None
(cid:18)(cid:1)(cid:16)(cid:18)(cid:5)(cid:19)(cid:25)(cid:4)(cid:5)(cid:1)(cid:19)(cid:9)(cid:16)(cid:4)(cid:1)(cid:18)(cid:10)(cid:5)(cid:19)(cid:10)(cid:4)
108.If a, b, c, d are in G.P. then the value of is ____
(A) 0 (B) 1 (C) –1 (D) None
(cid:1)(cid:2)(cid:19)(cid:26) (cid:4)(cid:5)(cid:6)(cid:6)(cid:5)(cid:7)(cid:8) (cid:9)(cid:10)(cid:5)(cid:11)(cid:12)(cid:4)(cid:12)(cid:13)(cid:7)(cid:4)(cid:14)(cid:8) (cid:15)(cid:13)(cid:16)(cid:15)
Copyright -The Institute of Chartered Accountants of India
(cid:1)(cid:16)(cid:18)(cid:9)(cid:18)(cid:19)(cid:9)(cid:19)(cid:25)(cid:4)(cid:10)(cid:5)(cid:1)(cid:16)(cid:10)(cid:9)(cid:18)(cid:10)(cid:9)(cid:19)(cid:10)(cid:4)(cid:1)(cid:18)(cid:10)(cid:9)(cid:19)(cid:10)(cid:9)(cid:25)(cid:10)(cid:4)
109.If a, b, c, d are in G.P. then the value of is _________.
(A) 0 (B) 1 (C) –1 (D) None
(cid:16)(cid:9)(cid:18)(cid:12) (cid:18)(cid:9)(cid:19)(cid:12) (cid:19)(cid:9)(cid:25)
110.If a, b, c, d are in G.P. then are in
(A) A.P. (B) G.P. (C) H.P. (D) None
(cid:16)(cid:10)(cid:9)(cid:18)(cid:10)(cid:12) (cid:16)(cid:18)(cid:9)(cid:18)(cid:19)(cid:12) (cid:18)(cid:10)(cid:9)(cid:19)(cid:10)
111.If a, b, c are in G.P. then are in
(A) A.P. (B) G.P. (C) H.P. (D) None
112.If a, b, x, y, z are positive numbers such that a, x, b are in A.P. and a, y, b are in G.P. and
"(cid:26)(cid:1)(cid:10)(cid:16)(cid:18)(cid:4)(cid:6)(cid:1)(cid:16)(cid:9)(cid:18)(cid:4)
then
! "
(A) x y z are in G.P. (B) ≥ ≥ (C) both (D) None
(cid:1)(cid:16)(cid:5)(cid:18)(cid:9)(cid:19)(cid:4)(cid:1)(cid:16)(cid:9)(cid:18)(cid:9)(cid:19)(cid:4)(cid:10)(cid:5)(cid:1)(cid:16)(cid:9)(cid:18)(cid:9)(cid:19)(cid:4)(cid:1)(cid:16)(cid:10)(cid:9)(cid:18)(cid:10)(cid:9)(cid:19)(cid:10)(cid:4)
113. If a, b, c are in G.P. then the value of is given by
(A) 0 (B) 1 (C) –1 (D) None
(cid:16)(cid:1)(cid:18)(cid:10)(cid:9)(cid:19)(cid:10)(cid:4)(cid:5)(cid:19)(cid:1)(cid:16)(cid:10)(cid:9)(cid:18)(cid:10)(cid:4)
114. If a, b, c are in G.P. then the value of is given by
(A) 0 (B) 1 (C) –1 (D) None
(cid:16)(cid:10)(cid:18)(cid:10)(cid:19)(cid:10)(cid:1)(cid:16)(cid:5)(cid:3)(cid:9)(cid:18)(cid:5)(cid:3)(cid:9)(cid:19)(cid:5)(cid:3)(cid:4)(cid:5)(cid:1)(cid:16)(cid:3)(cid:9)(cid:18)(cid:3)(cid:9)(cid:19)(cid:3)(cid:4)
115. If a, b, c are in G.P. then the value of is given by
(A) 0 (B) 1 (C) –1 (D) None
(cid:1)(cid:16)(cid:5)(cid:18)(cid:4)(cid:10)(cid:12) (cid:1)(cid:18)(cid:5)(cid:19)(cid:4)(cid:10)(cid:12) (cid:1)(cid:19)(cid:5)(cid:25)(cid:4)(cid:10)
116. If a, b, c, d are in G.P. then are in
(A) A.P. (B) G.P. (C) H.P. (D) None
(cid:1)(cid:18)(cid:5)(cid:19)(cid:4)(cid:10)(cid:9)(cid:1)(cid:19)(cid:5)(cid:16)(cid:4)(cid:10)(cid:9)(cid:1)(cid:25)(cid:5)(cid:18)(cid:4)(cid:10)(cid:5)(cid:1)(cid:16)(cid:5)(cid:25)(cid:4)(cid:10)
117. If a b c d are in G.P. then the value of is given by
(A) 0 (B) 1 (C) –1 (D) None
(cid:1)(cid:16)(cid:5)(cid:18)(cid:4)(cid:12)(cid:1)(cid:18)(cid:5)(cid:19)(cid:4)(cid:12)(cid:1)(cid:19)(cid:5)(cid:16)(cid:4) (cid:1)(cid:16)(cid:9)(cid:18)(cid:9)(cid:19)(cid:4)(cid:10)(cid:5)(cid:3)(cid:1)(cid:16)(cid:18)(cid:9)(cid:18)(cid:19)(cid:9)(cid:19)(cid:16)(cid:4)
118. If are in G.P. then the value of is given by
(A) 0 (B) 1 (C) –1 (D) None
119. If (cid:16)(cid:2)(cid:6) (cid:26)(cid:18)(cid:2)(cid:6)!(cid:26)(cid:19)(cid:2)(cid:6)" and a, b, c are in G.P. then x, y, z are in
(A) A.P. (B) G.P. (C) H.P. (D) None
120.If x = a + a/r + a/r2 + ..... α, y = b – b/r + b/r2 – ..... α, and z = c + c/r2 + c/r4 + .....
(cid:6)(cid:7) (cid:2)(cid:3)
(cid:4)
α, then the value of − is
(cid:5) (cid:1)
(A) 0 (B) 1 (C) –1 (D) None
(cid:6)(cid:17)(cid:15)(cid:18)(cid:16) (cid:1)(cid:2)(cid:19)(cid:25)
Copyright -The Institute of Chartered Accountants of India
(cid:1)(cid:2)(cid:3)(cid:4)(cid:2)(cid:5)(cid:6)(cid:2)(cid:7)(cid:8)(cid:5)(cid:9)(cid:7)(cid:1)(cid:2)(cid:10)(cid:11)(cid:2)(cid:1)(cid:12)(cid:8)(cid:10)(cid:11)(cid:13)(cid:14)(cid:15)(cid:2)(cid:13)(cid:11)(cid:6)(cid:7)(cid:8)(cid:5)(cid:9)(cid:7)(cid:16)(cid:2)(cid:17)(cid:15)(cid:2)(cid:13)(cid:10)(cid:11)(cid:6)(cid:7)(cid:18)(cid:10)(cid:17)(cid:16)(cid:10)(cid:2)(cid:1)(cid:1)(cid:11)(cid:17)(cid:5)(cid:1)
(cid:10)(cid:12) (cid:18)(cid:10)(cid:12) !(cid:10)
121.If a, b, c are in A.P. a, x, b are in G.P. and b, y, c are in G.P then are in
(A) A.P. (B) G.P. (C) H.P. (D) None
(cid:16)(cid:12) (cid:18)(cid:5)(cid:16)(cid:12) (cid:19)(cid:5)(cid:16) (cid:16)(cid:26)(cid:18)(cid:6)(cid:3)(cid:26)(cid:19)(cid:6)(cid:15)
122.If are in G.P. and then a, b, c are in
(A) A.P. (B) G.P. (C) H.P. (D) None
(cid:16)(cid:12) (cid:18)(cid:12)(cid:1)(cid:19)(cid:9)(cid:2)(cid:4)
123.If are in G.P. and a = (b–c) 2 then a, b, c are in
(A) A.P. (B) G.P. (C) H.P. (D) None
# (cid:12) # (cid:12) # (cid:12)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)#
124.If are the sums of infinite G.P.s whose first terms are 1, 2, 3 …..n and
(cid:2) (cid:10) (cid:3) (cid:7)
# (cid:9)# (cid:9)# (cid:9) (cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)#
whose common ratios are 1/2, 1/3, ……1/(n+1) then the value of is
(cid:2) (cid:10) (cid:3) (cid:7)
(A)
(cid:1)(cid:7)(cid:6)(cid:10)(cid:4) (cid:1)(cid:7)(cid:9)(cid:3)(cid:4)
(B)
(cid:1)(cid:7)(cid:6)(cid:10)(cid:4) (cid:1)(cid:7)(cid:9)(cid:10)(cid:4)
(C)
(cid:1)(cid:7)(cid:6)(cid:10)(cid:4) (cid:1)(cid:7)(cid:9)(cid:2)(cid:4)
(D)
(cid:7)(cid:10)(cid:6)(cid:10)
125. The G.P. whose 3rd and 6th terms are 1, –1/8 respectively is
(A) 4, –2, 1 ….. (B) 4, 2, 1 ……. (C) 4, –1, 1/4 …….. (D) None
126.In a G.P. if the (p+ q)th term is m and the (p – q)th term is n then the pth term is_________.
(A) (cid:1)$(cid:7)(cid:4)(cid:2)(cid:6)(cid:10) (B) $(cid:7) (C) (cid:1)$(cid:9)(cid:7)(cid:4) (D) (cid:1)$(cid:5)(cid:7)(cid:4)
(cid:2)(cid:6) (cid:3)(cid:9)(cid:2)(cid:9)(cid:3)(cid:6) (cid:3)(cid:9)(cid:11)(cid:11)(cid:11)(cid:11)
127.The sum of n terms of the series is
(cid:1)(cid:2)(cid:6)(cid:28)(cid:4) (cid:1)(cid:3)(cid:9) (cid:3)(cid:4) (cid:1)(cid:3)(cid:7)(cid:6)(cid:10)(cid:5)(cid:2)(cid:4)(cid:12) (cid:1)(cid:2)(cid:6)(cid:28)(cid:4) (cid:1) (cid:3)(cid:9)(cid:2)(cid:4) (cid:1)(cid:3)(cid:7)(cid:6)(cid:10)(cid:5)(cid:2)(cid:4)(cid:12)
(A) (B)
(cid:1)(cid:2)(cid:6)(cid:28)(cid:4) (cid:1)(cid:3)(cid:9) (cid:3)(cid:4) (cid:1)(cid:3)(cid:7)(cid:6)(cid:10)(cid:9)(cid:2)(cid:4)(cid:12)
(C) (D) None
128. The sum of n terms of the series 5/2 – 1 + 2/5 – …… is
(cid:1)(cid:2)(cid:6)(cid:2)(cid:17)(cid:4) (cid:1)(cid:15)(cid:7)(cid:9)(cid:10)(cid:7)(cid:4)(cid:6)(cid:15)(cid:7)(cid:5)(cid:10) (cid:1)(cid:2)(cid:6)(cid:2)(cid:17)(cid:4) (cid:1)(cid:15)(cid:7)(cid:5)(cid:10)(cid:7)(cid:4)(cid:6)(cid:15)(cid:7)(cid:5)(cid:10)
(A) (B) (C) both (D) None
129. The sum of n terms of the series 0.3 + 0.03 + 0.003 + …….. is
(cid:1)(cid:2)(cid:6)(cid:3)(cid:4)(cid:1)(cid:2)(cid:5)(cid:2)(cid:6)(cid:2)(cid:27)(cid:7)(cid:4) (cid:1)(cid:2)(cid:6)(cid:3)(cid:4)(cid:1)(cid:2)(cid:9)(cid:2)(cid:6)(cid:2)(cid:27)(cid:7)(cid:4)
(A) (B) (C) both (D) None
130. The sum of first eight terms of G.P. is five times the sum of the first four terms. The common
ratio is _______.
(A) (cid:10) (B) (cid:5) (cid:10) (C) both (D) None
131.If the sum of n terms of a G.P. with first term 1 and common ratio 1/2 is 1+127/128, the
value of n is _______.
(A) 8 (B) 5 (C) 3 (D) None
(cid:1)(cid:2)(cid:19)(cid:3) (cid:4)(cid:5)(cid:6)(cid:6)(cid:5)(cid:7)(cid:8) (cid:9)(cid:10)(cid:5)(cid:11)(cid:12)(cid:4)(cid:12)(cid:13)(cid:7)(cid:4)(cid:14)(cid:8) (cid:15)(cid:13)(cid:16)(cid:15)
Copyright -The Institute of Chartered Accountants of India
132.If the sum of n terms of a G.P. with last term 128 and common ratio 2 is 255, the value of
n is _________.
(A) 8 (B) 5 (C) 3 (D) None
133.How many terms of the G.P. 1, 4, 16 …. are to be taken to have their sum 341?
(A) 8 (B) 5 (C) 3 (D) None
134.The sum of n terms of the series 5 + 55 + 555 + …….. is
(cid:1)(cid:15)(cid:27)(cid:6)(cid:30)(cid:2)(cid:4) (cid:1)(cid:2)(cid:27)(cid:7)(cid:5)(cid:2)(cid:4)(cid:5)(cid:1)(cid:15)(cid:6)(cid:29)(cid:4)(cid:7) (cid:1)(cid:15)(cid:27)(cid:6)(cid:30)(cid:2)(cid:4) (cid:1)(cid:2)(cid:27)(cid:7)(cid:9)(cid:2)(cid:4)(cid:5)(cid:1)(cid:15)(cid:6)(cid:29)(cid:4)(cid:7)
(A) (B)
(cid:1)(cid:15)(cid:27)(cid:6)(cid:30)(cid:2)(cid:4) (cid:1)(cid:2)(cid:27)(cid:7)(cid:9)(cid:2)(cid:4)(cid:9)(cid:1)(cid:15)(cid:6)(cid:29)(cid:4)(cid:7)
(C) (D) None
135.The sum of n terms of the series 0.5 + 0.55 + 0.555 + ………. is
(cid:1)(cid:15)(cid:6)(cid:29)(cid:4)(cid:7)(cid:5)(cid:1)(cid:15)(cid:6)(cid:30)(cid:2)(cid:4)(cid:1)(cid:2)(cid:5)(cid:2)(cid:27)(cid:5)(cid:7)(cid:4) (cid:1)(cid:15)(cid:6)(cid:29)(cid:4)(cid:7)(cid:9)(cid:1)(cid:15)(cid:6)(cid:30)(cid:2)(cid:4)(cid:1)(cid:2)(cid:5)(cid:2)(cid:27)(cid:5)(cid:7)(cid:4)
(A) (B)
(cid:1)(cid:15)(cid:6)(cid:29)(cid:4)(cid:7)(cid:9)(cid:1)(cid:15)(cid:6)(cid:30)(cid:2)(cid:4)(cid:1)(cid:2)(cid:9)(cid:2)(cid:27)(cid:5)(cid:7)(cid:4)
(C) (D) None
(cid:2)(cid:11)(cid:27)(cid:3)(cid:9)(cid:2)(cid:11)(cid:27)(cid:3)(cid:10)(cid:9)(cid:2)(cid:11)(cid:27)(cid:3)(cid:3)(cid:9)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)
136. The sum of n terms of the series is
(cid:1)(cid:2)(cid:27)(cid:3)(cid:6)(cid:3)(cid:4)(cid:1)(cid:2)(cid:11)(cid:27)(cid:3)(cid:7)(cid:5)(cid:2)(cid:4) (cid:1)(cid:2)(cid:27)(cid:3)(cid:6)(cid:3)(cid:4)(cid:1)(cid:2)(cid:11)(cid:27)(cid:3)(cid:14) (cid:2)(cid:4) (cid:1)(cid:2)(cid:27)(cid:3)(cid:6)(cid:3)(cid:4)(cid:1)(cid:2)(cid:11)(cid:27)(cid:3)(cid:7)(cid:9)(cid:2)(cid:5)(cid:2)(cid:4)
(A) (B) + (C) (D) None
137. The sum upto infinity of the series 1/2 + 1/6 + 1/18 + …… is
(A) 3/4 (B) 1/4 (C) 1/2 (D) None
138. The sum upto infinity of the series 4 + 0.8 + 0.16 + …… is
(A) 5 (B) 10 (C) 8 (D) None
(cid:10)(cid:9)(cid:2)(cid:6) (cid:10)(cid:9)(cid:2)(cid:6)(cid:1)(cid:10) (cid:10)(cid:4)(cid:9)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)
139. The sum upto infinity of the series is
(A) (cid:10) (cid:10) (B) 2 (C) 4 (D) None
140. The sum upto infinity of the series 2/3 + 5/9 + 2/27 + 5/81 + ……. is
(A) 11/8 (B) 8/11 (C) 3/11 (D) None
(cid:1) (cid:10)(cid:9)(cid:2)(cid:4)(cid:9)(cid:2)(cid:9)(cid:1) (cid:10)(cid:5)(cid:2)(cid:4)(cid:9)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)
141. The sum upto infinity of the series is
(A) (cid:1)(cid:2)(cid:6)(cid:10)(cid:4)(cid:1)(cid:17)(cid:9)(cid:3) (cid:10)(cid:4) (B) (cid:1)(cid:2)(cid:6)(cid:10)(cid:4)(cid:1)(cid:17)(cid:5)(cid:3) (cid:10)(cid:4) (C) (cid:17)(cid:9)(cid:3) (cid:10) (D) None
(cid:1)(cid:2)(cid:9)(cid:10)(cid:5)(cid:10)(cid:4)(cid:9)(cid:1)(cid:10)(cid:5)(cid:2)(cid:9)(cid:10)(cid:5)(cid:17)(cid:4)(cid:9)(cid:1)(cid:10)(cid:5)(cid:10)(cid:9)(cid:10)(cid:5)(cid:28)(cid:4)(cid:9) (cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)
142.The sum upto infinity of the series is
(A) 7/3 (B) 3/7 (C) 4/7 (D) None
(cid:6)(cid:17)(cid:15)(cid:18)(cid:16) (cid:1)(cid:2)(cid:19)(cid:19)
Copyright -The Institute of Chartered Accountants of India
(cid:1)(cid:2)(cid:3)(cid:4)(cid:2)(cid:5)(cid:6)(cid:2)(cid:7)(cid:8)(cid:5)(cid:9)(cid:7)(cid:1)(cid:2)(cid:10)(cid:11)(cid:2)(cid:1)(cid:12)(cid:8)(cid:10)(cid:11)(cid:13)(cid:14)(cid:15)(cid:2)(cid:13)(cid:11)(cid:6)(cid:7)(cid:8)(cid:5)(cid:9)(cid:7)(cid:16)(cid:2)(cid:17)(cid:15)(cid:2)(cid:13)(cid:10)(cid:11)(cid:6)(cid:7)(cid:18)(cid:10)(cid:17)(cid:16)(cid:10)(cid:2)(cid:1)(cid:1)(cid:11)(cid:17)(cid:5)(cid:1)
(cid:17)(cid:6)(cid:14)(cid:5)(cid:15)(cid:6)(cid:14)(cid:10)(cid:9)(cid:17)(cid:6)(cid:14)(cid:3)(cid:5)(cid:15)(cid:6)(cid:14)(cid:17)(cid:9) (cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)
143.The sum upto infinity of the series is
(A) 23/48 (B) 25/48 (C) 1/2 (D) None
144.If the sum of infinite terms in a G.P. is 2 and the sum of their squares is 4/3 the series is
(A) 1, 1/2, 1/4 …… (B) 1, –1/2, 1/4 ……. (C) –1, –1/2, –1/4 …. (D) None
145.The infinite G.P. series with first term 1/4 and sum 1/3 is
(A) 1/4, 1/16, 1/64 … (B) 1/4, –1/16, 1/64 …(C) 1/4, 1/8, 1/16 …. (D) None
146.If the first term of a G.P. exceeds the second term by 2 and the sum to infinity is 50 the
series is __________.
(A) 10, 8, 32/5 … (B) 10, 8, 5/2 … (C) 10, 10/3, 10/9 …. (D) None
147.Three numbers in G.P. with their sum 130 and their product 27000 are _________.
(A) 10, 30, 90 … (B) 90, 30, 10 … (C) both (D) None
148.Three numbers in G.P. with their sum 13/3 and sum of their squares 91/9 are ____.
(A) 1/3, 1, 3 (B) 3, 1, 1/3 (C) both (D) None
149.Find five numbers in G.P. such that their product is 32 and the product of the last two is
108.
(A) 2/9, 2/3, 2, 6, 18 (B) 18, 6, 2, 2/3, 2/9 (C) both (D) None
150.If the continued product of three numbers in G.P. is 27 and the sum of their products in
pairs is 39 the numbers are _________.
(A) 1, 3, 9 (B) 9, 3, 1 (C) both (D) None
151.The numbers x, 8, y are in G.P. and the numbers x, y, –8 are in A.P. The values of x, y are
___________.
(A) 16, 4 (B) 4, 16 (C) both (D) None
(cid:1)(cid:2)(cid:19)(cid:20) (cid:4)(cid:5)(cid:6)(cid:6)(cid:5)(cid:7)(cid:8) (cid:9)(cid:10)(cid:5)(cid:11)(cid:12)(cid:4)(cid:12)(cid:13)(cid:7)(cid:4)(cid:14)(cid:8) (cid:15)(cid:13)(cid:16)(cid:15)
Copyright -The Institute of Chartered Accountants of India
AAAAANNNNNSSSSSWWWWWEEEEERRRRRSSSSS
1) C 31) A 61) A 91) A 121) A
2) A 32) A 62) A 92) A 122) A
3) B 33) A 63) A 93) A 123) A
4) A 34) B 64) A 94) C 124) A
5) A 35) B 65) A 95) C 125) A
6) B 36) A 66) A 96) B 126) A
7) C 37) B 67) A 97) B 127) A
8) C 38) C 68) A 98) B 128) C
9) A 39) D 69) A 99) A 129) A
10) A 40) D 70) A 100) C 130) C
11) B 41) A 71) A 101) B 131) A
12) B 42) A 72) A 102) A 132) A
13) A 43) D 73) A 103) A 133) B
14) C 44) D 74) A 104) A 134) A
15) B 45) C 75) A 105) C 135) A
16) A 46) A 76) A 106) C 136) A
17) A 47) B 77) B 107) A 137) A
18) A 48) C 78) A 108) A 138) A
19) A 49) A 79) A 109) A 139) A
20) C 50) B 80) A 110) B 140) A
21) B 51) A 81) A 111) B 141) A
22) A 52) D 82) A 112) C 142) A
23) A 53) B 83) A 113) A 143) A
24) A 54) B 84) A 114) A 144) A
25) C 55) A 85) A 115) A 145) A
26) A 56) A 86) A 116) B 146) A
27) C 57) D 87) A 117) A 147) C
28) C 58) A 88) B 118) A 148) C
29) A 59) A 89) C 119) A 149) A
30) C 60) A 90) D 120) A 150) C
151) A
(cid:6)(cid:17)(cid:15)(cid:18)(cid:16) (cid:1)(cid:2)(cid:19)(cid:21)
Copyright -The Institute of Chartered Accountants of India