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Sequences and Series

309 Questions Start JEE Mains Test
All Exams JEE Mains JEE Advanced TS-EAMCET AP-EAPCET BITSAT
2019 Q251 JEE Mains MCQ
14 Mar 2026
If the sum of the first 15 terms of the series ${\left( {{3 \over 4}} \right)^3} + {\left( {1{1 \over 2}} \right)^3} + {\left( {2{1 \over 4}} \right)^3} + {3^3} + {\left( {3{3 \over 4}} \right)^3} + ....$ is equal to 225 k, then k is equal to :
A.
9
B.
108
C.
27
D.
54
2019 Q252 JEE Mains MCQ
14 Mar 2026
If   nC4, nC5 and nC6 are in A.P., then n can be :
A.
11
B.
12
C.
9
D.
14
2019 Q253 JEE Mains MCQ
14 Mar 2026
Let  Sk = ${{1 + 2 + 3 + .... + k} \over k}.$ If   $S_1^2 + S_2^2 + .....\, + S_{10}^2 = {5 \over {12}}$A,  then A is equal to :
A.
283
B.
156
C.
301
D.
303
2019 Q254 JEE Mains MCQ
14 Mar 2026
The product of three consecutive terms of a G.P. is 512. If 4 is added to each of the first and the second of these terms, the three terms now form an A.P. Then the sum of the original three terms of the given G.P. is :
A.
36
B.
28
C.
32
D.
24
2019 Q255 JEE Mains MCQ
14 Mar 2026
Let x, y be positive real numbers and m, n positive integers. The maximum value of the expression ${{{x^m}{y^n}} \over {\left( {1 + {x^{2m}}} \right)\left( {1 + {y^{2n}}} \right)}}$ is :
A.
${1 \over 2}$
B.
${1 \over 4}$
C.
${{m + n} \over {6mn}}$
D.
1
2019 Q256 JEE Mains MCQ
14 Mar 2026
If 19th term of a non-zero A.P. is zero, then its (49th term) : (29th term) is :
A.
2 : 1
B.
4 : 1
C.
1 : 3
D.
3 : 1
2019 Q257 JEE Mains MCQ
14 Mar 2026
The sum of an infinite geometric series with positive terms is 3 and the sum of the cubes of its terms is ${{27} \over {19}}$.Then the common ratio of this series is :
A.
${4 \over 9}$
B.
${1 \over 3}$
C.
${2 \over 3}$
D.
${2 \over 9}$
2019 Q258 JEE Mains MCQ
14 Mar 2026
Let a1, a2, . . . . . ., a10 be a G.P.    If ${{{a_3}} \over {{a_1}}} = 25,$ then ${{{a_9}} \over {{a_5}}}$ equals
A.
53
B.
2(52)
C.
4(52)
D.
54
2019 Q259 JEE Mains MCQ
14 Mar 2026
Let a1, a2, a3, ..... a10 be in G.P. with ai > 0 for i = 1, 2, ….., 10 and S be the set of pairs (r, k), r, k $ \in $ N (the set of natural numbers) for which

$\left| {\matrix{ {{{\log }_e}\,{a_1}^r{a_2}^k} & {{{\log }_e}\,{a_2}^r{a_3}^k} & {{{\log }_e}\,{a_3}^r{a_4}^k} \cr {{{\log }_e}\,{a_4}^r{a_5}^k} & {{{\log }_e}\,{a_5}^r{a_6}^k} & {{{\log }_e}\,{a_6}^r{a_7}^k} \cr {{{\log }_e}\,{a_7}^r{a_8}^k} & {{{\log }_e}\,{a_8}^r{a_9}^k} & {{{\log }_e}\,{a_9}^r{a_{10}}^k} \cr } } \right|$ $=$ 0.

Then the number of elements in S, is -
A.
10
B.
4
C.
2
D.
infinitely many
2019 Q260 JEE Mains MCQ
14 Mar 2026
The sum of all two digit positive numbers which when divided by 7 yield 2 or 5 as remainder is -
A.
1356
B.
1256
C.
1365
D.
1465
2019 Q261 JEE Mains MCQ
14 Mar 2026
The sum of the following series

$1 + 6 + {{9\left( {{1^2} + {2^2} + {3^2}} \right)} \over 7} + {{12\left( {{1^2} + {2^2} + {3^2} + {4^2}} \right)} \over 9}$

       $ + {{15\left( {{1^2} + {2^2} + ... + {5^2}} \right)} \over {11}} + .....$ up to 15 terms, is :
A.
7520
B.
7510
C.
7830
D.
7820
2019 Q262 JEE Mains MCQ
14 Mar 2026
Let a, b and c be the 7th, 11th and 13th terms respectively of a non-constant A.P. If these are also three consecutive terms of a G.P., then ${a \over c}$ equal to :
A.
2
B.
${1 \over 2}$
C.
${7 \over 13}$
D.
4
2019 Q263 JEE Mains MCQ
14 Mar 2026
If a, b, c be three distinct real numbers in G.P. and a + b + c = xb , then x cannot be
A.
2
B.
-3
C.
4
D.
-2
2019 Q264 JEE Mains MCQ
14 Mar 2026
Let ${a_1},{a_2},.......,{a_{30}}$ be an A.P.,

$S = \sum\limits_{i = 1}^{30} {{a_i}} $ and $T = \sum\limits_{i = 1}^{15} {{a_{\left( {2i - 1} \right)}}} $.

If $a_5$ = 27 and S - 2T = 75, then $a_{10}$ is equal to :
A.
47
B.
42
C.
52
D.
57
2018 Q265 JEE Mains MCQ
14 Mar 2026
Let ${1 \over {{x_1}}},{1 \over {{x_2}}},...,{1 \over {{x_n}}}\,\,$ (xi $ \ne $ 0 for i = 1, 2, ..., n) be in A.P. such that x1=4 and x21 = 20. If n is the least positive integer for which ${x_n} > 50,$ then $\sum\limits_{i = 1}^n {\left( {{1 \over {{x_i}}}} \right)} $ is equal to :
A.
${1 \over 8}$
B.
3
C.
${{13} \over 8}$
D.
${{13} \over 4}$
2018 Q266 JEE Mains MCQ
14 Mar 2026
The sum of the first 20 terms of the series

$1 + {3 \over 2} + {7 \over 4} + {{15} \over 8} + {{31} \over {16}} + ...,$ is :
A.
$38 + {1 \over {{2^{19}}}}$
B.
$38 + {1 \over {{2^{20}}}}$
C.
$39 + {1 \over {{2^{20}}}}$
D.
$39 + {1 \over {{2^{19}}}}$
2018 Q267 JEE Mains MCQ
14 Mar 2026
Let A be the sum of the first 20 terms and B be the sum of the first 40 terms of the series
12 + 2.22 + 32 + 2.42 + 52 + 2.62 ...........
If B - 2A = 100$\lambda $, then $\lambda $ is equal to
A.
496
B.
232
C.
248
D.
464
2018 Q268 JEE Mains MCQ
14 Mar 2026
Let ${a_1}$, ${a_2}$, ${a_3}$, ......... ,${a_{49}}$ be in A.P. such that

$\sum\limits_{k = 0}^{12} {{a_{4k + 1}}} = 416$ and ${a_9} + {a_{43}} = 66$.

$a_1^2 + a_2^2 + ....... + a_{17}^2 = 140m$, then m is equal to
A.
33
B.
66
C.
68
D.
34
2018 Q269 JEE Mains MCQ
14 Mar 2026
Let    An = $\left( {{3 \over 4}} \right) - {\left( {{3 \over 4}} \right)^2} + {\left( {{3 \over 4}} \right)^3}$ $-$. . . . . + ($-$1)n-1 ${\left( {{3 \over 4}} \right)^n}$    and    Bn = 1 $-$ An.
Then, the least dd natural numbr p, so that Bn > An , for all n$ \ge $ p, is :
A.
9
B.
7
C.
11
D.
5
2018 Q270 JEE Mains MCQ
14 Mar 2026
If  a,   b,   c  are in A.P. and  a2,  b2,  c2 are in G.P. such that
a < b < c and   a + b + c = ${3 \over 4},$ then the value of a is :
A.
${1 \over 4} - {1 \over {4\sqrt 2 }}$
B.
${1 \over 4} - {1 \over {3\sqrt 2 }}$
C.
${1 \over 4} - {1 \over {2\sqrt 2 }}$
D.
${1 \over 4} - {1 \over {\sqrt 2 }}$
2018 Q271 JEE Mains MCQ
14 Mar 2026
If x1, x2, . . ., xn and ${1 \over {{h_1}}}$, ${1 \over {{h_2}}}$, . . . , ${1 \over {{h_n}}}$ are two A.P..s such that x3 = h2 = 8 and x8 = h7 = 20, then x5.h10 equals :
A.
2560
B.
2650
C.
3200
D.
1600
2018 Q272 JEE Mains MCQ
14 Mar 2026
If b is the first term of an infinite G.P. whose sum is five, then b lies in the interval :
A.
($-$ $\infty $, $-$10]
B.
($-$10, 0)
C.
(0, 10)
D.
[10, $\infty $)
2017 Q273 JEE Mains MCQ
14 Mar 2026
Let

Sn = ${1 \over {{1^3}}}$$ + {{1 + 2} \over {{1^3} + {2^3}}} + {{1 + 2 + 3} \over {{1^3} + {2^3} + {3^3}}} + ......... + {{1 + 2 + ....... + n} \over {{1^3} + {2^3} + ...... + {n^3}}}.$

If 100 Sn = n, then n is equal to :
A.
199
B.
99
C.
200
D.
19
2017 Q274 JEE Mains MCQ
14 Mar 2026
If three positive numbers a, b and c are in A.P. such that abc = 8, then the minimum possible value of b is :
A.
2
B.
4${^{{1 \over 3}}}$
C.
4${^{{2 \over 3}}}$
D.
4
2017 Q275 JEE Mains MCQ
14 Mar 2026
If the sum of the first n terms of the series $\,\sqrt 3 + \sqrt {75} + \sqrt {243} + \sqrt {507} + ......$ is $435\sqrt 3 ,$ then n equals :
A.
18
B.
15
C.
13
D.
29
2017 Q276 JEE Mains MCQ
14 Mar 2026
If the arithmetic mean of two numbers a and b, a > b > 0, is five times their geometric mean, then ${{a + b} \over {a - b}}$ is equal to :
A.
${{\sqrt 6 } \over 2}$
B.
${{3\sqrt 2 } \over 4}$
C.
${{7\sqrt 3 } \over {12}}$
D.
${{5\sqrt 6 } \over {12}}$
2017 Q277 JEE Mains MCQ
14 Mar 2026
For any three positive real numbers a, b and c,

9(25${a^2}$ + b2) + 25(c2 - 3$a$c) = 15b(3$a$ + c).
Then
A.
b, c and $a$ are in G.P.
B.
b, c and $a$ are in A.P.
C.
$a$, b and c are in A.P.
D.
$a$, b and c are in G.P.
2016 Q278 JEE Mains MCQ
14 Mar 2026
If   A > 0, B > 0   and    A + B = ${\pi \over 6}$,

then the minimum value of tanA + tanB is :
A.
$\sqrt 3 - \sqrt 2 $
B.
$2 - \sqrt 3 $
C.
$4 - 2\sqrt 3 $
D.
${2 \over {\sqrt 3 }}$
2016 Q279 JEE Mains MCQ
14 Mar 2026
Let z = 1 + ai be a complex number, a > 0, such that z3 is a real number.

Then the sum 1 + z + z2 + . . . . .+ z11 is equal to :
A.
$ - 1250\,\sqrt 3 \,i$
B.
$ 1250\,\sqrt 3 \,i$
C.
$1365\,\sqrt 3 i$
D.
$-$ $1365\,\sqrt 3 i$
2016 Q280 JEE Mains MCQ
14 Mar 2026
Let a1, a2, a3, . . . . . . . , an, . . . . . be in A.P.

If a3 + a7 + a11 + a15 = 72,

then the sum of its first 17 terms is equal to :
A.
306
B.
153
C.
612
D.
204
2016 Q281 JEE Mains MCQ
14 Mar 2026
Let x, y, z be positive real numbers such that x + y + z = 12 and x3y4z5 = (0.1) (600)3. Then x3 + y3 + z3is equal to :
A.
270
B.
258
C.
342
D.
216
2016 Q282 JEE Mains MCQ
14 Mar 2026
If the ${2^{nd}},{5^{th}}\,and\,{9^{th}}$ terms of a non-constant A.P. are in G.P., then the common ratio of this G.P. is :
A.
1
B.
${7 \over 4}$
C.
${8 \over 5}$
D.
${4 \over 3}$
2016 Q283 JEE Mains MCQ
14 Mar 2026
If the sum of the first ten terms of the series ${\left( {1{3 \over 5}} \right)^2} + {\left( {2{2 \over 5}} \right)^2} + {\left( {3{1 \over 5}} \right)^2} + {4^2} + {\left( {4{4 \over 5}} \right)^2} + .......is\,{{16} \over 5}m,$ then m is equal to :
A.
100
B.
99
C.
102
D.
101
2015 Q284 JEE Mains MCQ
14 Mar 2026
The sum of first 9 terms of the series.

${{{1^3}} \over 1} + {{{1^3} + {2^3}} \over {1 + 3}} + {{{1^3} + {2^3} + {3^3}} \over {1 + 3 + 5}} + ......$
A.
142
B.
192
C.
71
D.
96
2015 Q285 JEE Mains MCQ
14 Mar 2026
If m is the A.M. of two distinct real numbers l and n $(l,n > 1)$ and ${G_1},{G_2}$ and ${G_3}$ are three geometric means between $l$ and n, then $G_1^4\, + 2G_2^4\, + G_3^4$ equals:
A.
$4\,lm{n^2}$
B.
$4\,{l^2}{m^2}{n^2}$
C.
$4\,{l^2}m\,n$
D.
$4\,l\,{m^2}n$
2014 Q286 JEE Mains MCQ
14 Mar 2026
Three positive numbers form an increasing G.P. If the middle term in this G.P. is doubled, the new numbers are in A.P. then the common ratio of the G.P. is :
A.
$2 - \sqrt 3 $
B.
$2 + \sqrt 3 $
C.
$\sqrt 2 + \sqrt 3 $
D.
$3 + \sqrt 2 $
2014 Q287 JEE Mains MCQ
14 Mar 2026
If ${(10)^9} + 2{(11)^1}\,({10^8}) + 3{(11)^2}\,{(10)^7} + ......... + 10{(11)^9} = k{(10)^9},$, then k is equal to :
A.
100
B.
110
C.
${{121} \over {10}}$
D.
${{441} \over {100}}$
2013 Q288 JEE Mains MCQ
14 Mar 2026
The sum of first 20 terms of the sequence 0.7, 0.77, 0.777,........,is
A.
${7 \over {81}}\left( {179 - {{10}^{ - 20}}} \right)$
B.
$\,{7 \over 9}\left( {99 - {{10}^{ - 20}}} \right)$
C.
${7 \over {81}}\left( {179 + {{10}^{ - 20}}} \right)$
D.
${7 \over 9}\left( {99 + {{10}^{ - 20}}} \right)$
2012 Q289 JEE Mains MCQ
14 Mar 2026

Statement-1: The sum of the series 1 + (1 + 2 + 4) + (4 + 6 + 9) + (9 + 12 + 16) +.....+ (361 + 380 + 400) is 8000.

Statement-2: $\sum\limits_{k = 1}^n {\left( {{k^3} - {{(k - 1)}^3}} \right)} = {n^3}$, for any natural number n.

A.
Statement-1 is false, Statement-2 is true.
B.
Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
C.
Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
D.
Statement-1 is true, Statement-2 is false.
2011 Q290 JEE Mains MCQ
14 Mar 2026
A man saves ₹ 200 in each of the first three months of his service. In each of the subsequent months his saving increases by ₹ 40 more than the saving of immediately previous month. His total saving from the start of service will be ₹ 11040 after
A.
19 months
B.
20 months
C.
21 months
D.
18 months
2010 Q291 JEE Mains MCQ
14 Mar 2026
A person is to count 4500 currency notes. Let ${a_n}$ denote the number of notes he counts in the ${n^{th}}$ minute. If ${a_1}$ = ${a_2}$ = ....= ${a_{10}}$= 150 and ${a_{10}}$, ${a_{11}}$,.... are in an AP with common difference - 2, then the time taken by him to count all notes is
A.
34 minutes
B.
125 minutes
C.
135 minutes
D.
24 minutes
2009 Q292 JEE Mains MCQ
14 Mar 2026
The sum to infinite term of the series $1 + {2 \over 3} + {6 \over {{3^2}}} + {{10} \over {{3^3}}} + {{14} \over {{3^4}}} + .....$ is
A.
3
B.
4
C.
6
D.
2
2008 Q293 JEE Mains MCQ
14 Mar 2026
The first two terms of a geometric progression add up to 12. the sum of the third and the fourth terms is 48. If the terms of the geometric progression are alternately positive and negative, then the first term is
A.
- 4
B.
- 12
C.
12
D.
4
2007 Q294 JEE Mains MCQ
14 Mar 2026
In a geometric progression consisting of positive terms, each term equals the sum of the next two terns. Then the common ratio of its progression is equals
A.
${\sqrt 5 }$
B.
$\,{1 \over 2}\left( {\sqrt 5 - 1} \right)$
C.
${1 \over 2}\left( {1 - \sqrt 5 } \right)$
D.
${1 \over 2}\sqrt 5 $.
2007 Q295 JEE Mains MCQ
14 Mar 2026
The sum of series ${1 \over {2!}} - {1 \over {3!}} + {1 \over {4!}} - .......$ upto infinity is
A.
${e^{ - {1 \over 2}}}$
B.
${e^{ + {1 \over 2}}}$
C.
${e^{ - 2}}$
D.
${e^{ - 1}}$
2006 Q296 JEE Mains MCQ
14 Mar 2026
If ${{a_1},{a_2},....{a_n}}$ are in H.P., then the expression ${{a_1}\,{a_2} + \,{a_2}\,{a_3}\, + .... + {a_{n - 1}}\,{a_n}}$ is equal to
A.
$n({a_1}\, - {a_n})$
B.
$(n - 1)({a_1}\, - {a_n})$
C.
$n{a_1}{a_n}$
D.
$(n - 1)\,\,{a_1}{a_n}$
2006 Q297 JEE Mains MCQ
14 Mar 2026
Let ${a_1}$, ${a_2}$, ${a_3}$.....be terms on A.P. If ${{{a_1} + {a_2} + .....{a_p}} \over {{a_1} + {a_2} + .....{a_q}}} = {{{p^2}} \over {{q^2}}},\,p \ne q,\,then\,{{{a_6}} \over {{a_{21}}}}\,$ equals
A.
${{41} \over {11}}$
B.
${7 \over 2}$
C.
${2 \over 7}$
D.
${{11} \over {41}}$
2005 Q298 JEE Mains MCQ
14 Mar 2026
The sum of the series $1 + {1 \over {4.2!}} + {1 \over {16.4!}} + {1 \over {64.6!}} + .......$ ad inf. is
A.
${{e - 1} \over {\sqrt e }}\,$
B.
${{e + 1} \over {\sqrt e }}$
C.
${{e - 1} \over {2\sqrt e }}$
D.
${{e + 1} \over {2\sqrt e }}$
2005 Q299 JEE Mains MCQ
14 Mar 2026
If $x = \sum\limits_{n = 0}^\infty {{a^n},\,\,y = \sum\limits_{n = 0}^\infty {{b^n},\,\,z = \sum\limits_{n = 0}^\infty {{c^n},} } } \,\,$ where a, b, c are in A.P and $\,\left| a \right| < 1,\,\left| b \right| < 1,\,\left| c \right| < 1$ then x, y, z are in
A.
G.P.
B.
A.P.
C.
Arithmetic-Geometric Progression
D.
H.P.
2004 Q300 JEE Mains MCQ
14 Mar 2026
The sum of series ${1 \over {2\,!}} + {1 \over {4\,!}} + {1 \over {6\,!}} + ........$ is
A.
${{\left( {{e^2} - 2} \right)} \over e}\,$
B.
${{{{\left( {e - 1} \right)}^2}} \over {2e}}$
C.
${{\left( {{e^2} - 1} \right)} \over {2e}}\,$
D.
${{\left( {{e^2} - 1} \right)} \over 2}$