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Sequences and Series

462 Questions Start JEE Mains Test
All Exams JEE Mains JEE Advanced TS-EAMCET AP-EAPCET BITSAT
2007 Q401 JEE Advanced MCQ
14 Mar 2026

The sum V$_1$ + V$_2$ + ... + V$_n$ is

A.
${1 \over {12}}n(n + 1)(3{n^2} - n + 1)$
B.
${1 \over {12}}n(n + 1)(3{n^2} + n + 2)$
C.
${1 \over 2}n(2{n^2} - n + 1)$
D.
${1 \over 3}(2{n^3} - 2n + 3)$
2007 Q402 JEE Advanced MCQ
14 Mar 2026

T$_r$ is always

A.
an odd number
B.
an even number
C.
a prime number
D.
a composite number
2007 Q403 JEE Advanced MCQ
14 Mar 2026

Which one of the following is a correct statement?

A.
Q$_1$, Q$_2$, Q$_3$, ... are in A.P. with common difference 5
B.
Q$_1$, Q$_2$, Q$_3$, ... are in A.P. with common difference 6
C.
Q$_1$, Q$_2$, Q$_3$, ... are in A.P. with common difference 11
D.
Q$_1$ = Q$_2$ = Q$_3$, ...
2006 Q404 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 Q405 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 Q406 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 Q407 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.
2005 Q408 JEE Advanced MCQ
14 Mar 2026
In the quadratic equation $\,\,a{x^2} + bx + c = 0,$ $\Delta $ $ = {b^2} - 4ac$ and $\alpha + \beta ,\,{\alpha ^2} + {\beta ^2},\,{\alpha ^3} + {\beta ^3},$ are in G.P. where $\alpha ,\beta $ are the root of $\,\,a{x^2} + bx + c = 0,$ then
A.
$\Delta \ne 0$
B.
$b\Delta = 0$
C.
$c\Delta = 0$
D.
$\Delta = 0$
2005 Q409 JEE Advanced MCQ
14 Mar 2026

If total number of runs scored in $n$ matches is $\left(\frac{n+1}{4}\right)\left(2^{n+1}-n-2\right)$ where $n > 1$, and the runs scored in the $k^{\text {th }}$ match are given by $k .2^{n+1-k}$, where $1 \leq k \leq n$. Find, $n$.

A.
5
B.
7
C.
15
D.
1
2004 Q410 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.
${{\left( {{e^2} - 1} \right)} \over 2}$
2004 Q411 JEE Mains MCQ
14 Mar 2026
Let ${{T_r}}$ be the rth term of an A.P. whose first term is a and common difference is d. If for some positive integers m, n, $m \ne n,\,\,{T_m} = {1 \over n}\,\,and\,{T_n} = {1 \over m},\,$ then a - d equals
A.
${1 \over m} + {1 \over n}$
B.
1
C.
${1 \over {m\,n}}$
D.
0
2004 Q412 JEE Mains MCQ
14 Mar 2026
The sum of the first n terms of the series ${1^2} + {2.2^2} + {3^2} + {2.4^2} + {5^2} + {2.6^2} + ....\,is\,{{n{{(n + 1)}^2}} \over 2}$ when n is even. When n is odd the sum is
A.
${\left[ {{{n(n + 1)} \over 2}} \right]^2}$
B.
${{{n^2}(n + 1)} \over 2}$
C.
${{n{{(n + 1)}^2}} \over 4}$
D.
$\,{{3n(n + 1)} \over 2}$
2004 Q413 JEE Advanced MCQ
14 Mar 2026
An infinite G.P. has first term '$x$' and sum '$5$', then $x$ belongs to
A.
$x < - 10$
B.
$ - 10 < x < 0$
C.
$0 < x < 10$
D.
$x > 10$
2003 Q414 JEE Mains MCQ
14 Mar 2026
The sum of the serier ${1 \over {1.2}} - {1 \over {2.3}} + {1 \over {3.4}}..............$ up to $\infty $ is equal to
A.
$\log {\,_e}\left( {{4 \over e}} \right)\,\,$
B.
$2\,\log {\,_e}2$
C.
$\log {\,_e}2 - 1\,$
D.
$\log {\,_e}2$
2003 Q415 JEE Advanced Numerical
14 Mar 2026
If a, b, c are in A.P., ${a^2}$, ${b^2}$, ${c^2}$ are in H.P., then prove that either a = b = c or a, b, ${ - {c \over 2}}$ form a G.P.
2002 Q416 JEE Mains MCQ
14 Mar 2026
The value of $\,{2^{1/4}}.\,\,{4^{1/8}}.\,{8^{1/16}}...\infty $ is
A.
1
B.
2
C.
3/2
D.
4
2002 Q417 JEE Mains MCQ
14 Mar 2026
${1^3} - \,\,{2^3} + {3^3} - {4^3} + ... + {9^3} = $
A.
425
B.
- 425
C.
475
D.
- 475
2002 Q418 JEE Mains MCQ
14 Mar 2026
l, m, n are the ${p^{th}}$, ${q^{th}}$ and ${r^{th}}$ term of a G.P all positive, $then\,\left| {\matrix{ {\log \,l} & p & 1 \cr {\log \,m} & q & 1 \cr {\log \,n} & r & 1 \cr } } \right|\,equals$
A.
- 1
B.
2
C.
1
D.
0
2002 Q419 JEE Mains MCQ
14 Mar 2026
Sum of infinite number of terms of GP is 20 and sum of their square is 100. The common ratio of GP is
A.
5
B.
3/5
C.
8/5
D.
1/5
2002 Q420 JEE Mains MCQ
14 Mar 2026
If 1, ${\log _9}\,\,({3^{1 - x}} + 2),\,\,{\log _3}\,\,({4.3^x} - 1)$ are in A.P. then x equals
A.
${\log _3}\,4\,\,\,$
B.
$1 - \,{\log _3}\,4\,$
C.
$1 - \,{\log _4}\,3$
D.
${\log _4}\,3$
2002 Q421 JEE Mains MCQ
14 Mar 2026
Fifth term of a GP is 2, then the product of its 9 terms is
A.
256
B.
512
C.
1024
D.
none of these
2002 Q422 JEE Advanced MCQ
14 Mar 2026
Suppose $a, b, c$ are in A.P. and ${a^2},{b^2},{c^2}$ are in G.P. If $a < b < c$ and $a + b + c = {3 \over 2},$ then the value of $a$ is
A.
${1 \over {2\sqrt 2 }}$
B.
${1 \over {2\sqrt 3 }}$
C.
${1 \over 2} - {1 \over {\sqrt 3 }}$
D.
${1 \over 2} - {1 \over {\sqrt 2 }}$
2002 Q423 JEE Advanced Numerical
14 Mar 2026
Let a, b be positive real numbers. If a, ${{A_1},{A_2}}$, b are in arithmetic progression, a, ${{G_1},{G_2}}$, b are in geometric progression and a, ${{H_1},{H_2}}$, b are in harmonic progression, show that $\,{{{G_1},{G_2}} \over {{H_1},{H_2}}} = {{{A_1} + {A_2}} \over {{H_1} + {H_2}}} = {{(2a + b)\,(a + 2b)} \over {9ab}}$.
2001 Q424 JEE Advanced MCQ
14 Mar 2026
Let the positive numbers $a,b,c,d$ be in A.P. Then $abc,$ $abd,$ $acd,$ $bcd,$ are
A.
NOT in A.P./GP./H.P.
B.
inA.P.
C.
in GP.
D.
in H.P.
2001 Q425 JEE Advanced MCQ
14 Mar 2026
If the sum of the first $2n$ terms of the A.P.$2,5,8,......,$ is equal to the sum of the first $n$ terms of the A.P.$57,59,61,.....,$ then $n$ equals
A.
10
B.
12
C.
11
D.
13
2001 Q426 JEE Advanced MCQ
14 Mar 2026
Let $\alpha $, $\beta $ be the roots of ${x^2} - x + p = 0$ and $\gamma ,\delta $ be the roots of ${x^2} - 4x + q = 0.$ If $\alpha ,\beta ,\gamma ,\delta $ are in G.P., then the integral values of $p$ and $q$ respectively, are
A.
$-2,-32$
B.
$-2,3$
C.
$-6,3$
D.
$-6,-32$
2001 Q427 JEE Advanced Numerical
14 Mar 2026
Let ${a_1}$, ${a_2}$,.....,${a_n}$ be positive real numbers in geometric progression. For each n, let ${A_n}$, ${G_n}$, ${H_n}$ be respectively, the arithmetic mean , geometric mean, and harmonic mean of ${a_1}$,${a_2}$......,${a_n}$. Find an expression for the geometric mean of ${G_1}$,${G_2}$,.....,${G_n}$ in terms of ${A_1}$,${A_2}$,.....,${A_n}$,${H_n}$,${H_1}$,${H_2}$,........,${H_n}$.
2000 Q428 JEE Advanced MCQ
14 Mar 2026
Consider an infinite geometric series with first term a and common ratio $r$. If its sum is 4 and the second term is 3/4, then
A.
$a = {4 \over 7},r = {3 \over 7}\,\,\,\,$
B.
$a = 2,\,r = {3 \over 8}$
C.
$a = {3 \over 2},r = {1 \over 2}$
D.
$a = 3,\,r = {1 \over 4}$
2000 Q429 JEE Advanced Numerical
14 Mar 2026
The fourth power of the common difference of an arithmatic progression with integer entries is added to the product of any four consecutive terms of it. Prove that the resulting sum is the square of an integer.
1999 Q430 JEE Advanced MCQ
14 Mar 2026
The harmonic mean of the roots of the equation $\left( {5 + \sqrt 2 } \right){x^2} - \left( {4 + \sqrt 5 } \right)x + 8 + 2\sqrt 5 = 0$ is
A.
2
B.
4
C.
6
D.
8
1999 Q431 JEE Advanced MCQ
14 Mar 2026
Let ${a_1},{a_2},......{a_{10}}$ be in $A,\,P,$ and ${h_1},{h_2},......{h_{10}}$ be in H.P. If ${a_1} = {h_1} = 2$ and ${a_{10}} = {h_{10}} = 3,$ then ${a_4}{h_7}$ is
A.
2
B.
3
C.
5
D.
6
1999 Q432 JEE Advanced MSQ
14 Mar 2026
For a positive integer $n$, let
$a\left( n \right) = 1 + {1 \over 2} + {1 \over 3} + {1 \over 4} + .....\,{1 \over {\left( {{2^n}} \right) - 1}}$. Then
A.
$a\left( {100} \right) \le 100$
B.
$a\left( {100} \right) > 100$
C.
$a\left( {200} \right) \le 100$
D.
$a\left( {200} \right) > 100$
1999 Q433 JEE Advanced Numerical
14 Mar 2026
Let a, b, c, d be real numbers in G.P. If u, v, w, satisfy the system of equations
u + 2v + 3w = 6
4u + 5v + 6w = 12
6u + 9v = 4

then show that the roots of the equation $\left( {{1 \over u} + {1 \over v} + {1 \over w}} \right){x^2}$
$ + [{(b - c)^2} + {(c - a)^2} + {(d - b)^2}]x + u + v + w = 0$ and $20{x^2} + 10{(a - d)^2}x - 9 = 0$ are reciprocals of each other.

1998 Q434 JEE Advanced MCQ
14 Mar 2026
Let $n$ be an odd integer. If $\sin n\theta = \sum\limits_{r = 0}^n {{b_r}{{\sin }^r}\theta ,} $ for every value of $\theta ,$ then
A.
${b_0} = 1,\,b = 3$
B.
${b_0} = 0,\,{b_1} = n$
C.
${b_0} = - 1,\,{b_1} = n$
D.
${b_0} = 0,\,{b_1} = {n^2} - 3n + 3$
1998 Q435 JEE Advanced MCQ
14 Mar 2026
If $x > 1,y > 1,z > 1$ are in G.P., then ${1 \over {1 + In\,x}},{1 \over {1 + In\,y}},{1 \over {1 + In\,z}}$ are in
A.
A.P.
B.
H.P.
C.
G.P.
D.
None of these
1998 Q436 JEE Advanced MCQ
14 Mar 2026
Let ${T_r}$ be the ${r^{th}}$ term of an A.P., for $r=1, 2, 3, ....$ If for some positive integers $m$, $n$ we have
${T_m} = {1 \over n}$ and ${T_n} = {1 \over m},$ then ${T_n} = {1 \over m},$ equals
A.
${1 \over {mn}}$
B.
${1 \over {mn}} + {1 \over n}$
C.
$1$
D.
$0$
1997 Q437 JEE Advanced Numerical
14 Mar 2026
Let $p$ and $q$ be roots of the equation ${x^2} - 2x + A = 0$ and let $r$ and $s$ be the roots of the equation ${x^2} - 18x + B = 0.$ If $p < q < r < s$ are in arithmetic progression, then $A = \,..........$ and $B = \,..........$
1996 Q438 JEE Advanced Numerical
14 Mar 2026
The real numbers ${x_1}$, ${x_2}$, ${x_3}$ satisfying the equation ${x^3} - {x^2} + \beta x + \gamma = 0$ are in AP. Find the intervals in which $\beta \,\,and\,\gamma $ lie.
1996 Q439 JEE Advanced Numerical
14 Mar 2026
For any odd integer $n$ $ \ge 1,\,\,{n^3} - {\left( {n - 1} \right)^3} + .... + {\left( { - 1} \right)^{n - 1}}\,{1^3} = ........$
1994 Q440 JEE Advanced MCQ
14 Mar 2026
If $In\left( {a + c} \right),In\left( {a - c} \right),In\left( {a - 2b + c} \right)$ are in A.P., then
A.
$a,\,b,\,c$ are in A.P.
B.
${a^2},\,{b^2},\,{c^2}$ are in A.P.
C.
$a,\,b,\,c$ are in G.P.
D.
$a,\,b,\,c$ are in H.P.
1993 Q441 JEE Advanced MSQ
14 Mar 2026
For $0 < \phi < \pi /2,$ if
$x = $$\sum\limits_{n = 0}^\infty {{{\cos }^{2n}}\phi ,y = \sum\limits_{n = 0}^\infty {{{\sin }^{2n}}\phi ,\,\,\,\,z = \sum\limits_{n = 0}^{} {{{\cos }^{2n}}\phi {{\sin }^{2n}}\phi } } } \infty $ then
A.
$xyz = xz + y$
B.
$xyz = xy + z$
C.
$xyz = x + y + z$
D.
$xyz = yz + x$
1992 Q442 JEE Advanced Numerical
14 Mar 2026
Let the harmonic mean and geometric mean of two positive numbers be the ratio 4 : 5. Then the two number are in the ratio .........
1991 Q443 JEE Advanced Numerical
14 Mar 2026
If ${S_1}$, ${S_2}$, ${S_3}$,.............,${S_n}$ are the sums of infinite geometric series whose first terms are 1, 2, 3, ...................,n and whose common ratios are ${1 \over 2}$, ${1 \over 3}$, ${1 \over 4}$,....................$\,{1 \over {n + 1}}$ respectively, then find the values of ${S_1}^2 + {S_2}^2 + {S_3}^2 + ....... + {S^2}_{2n - 1}$
1991 Q444 JEE Advanced Numerical
14 Mar 2026
Let p be the first of the n arithmetic means between two numbers and q the first of n harmonic means between the same numbers. Show that q does not lie between p and $\,{\left( {{{n + 1} \over {n - 1}}} \right)^2}\,p$.
1990 Q445 JEE Advanced MCQ
14 Mar 2026
The number ${\log _2}\,7$ is
A.
an integer
B.
a rational number
C.
an irrational number
D.
a prime number
1990 Q446 JEE Advanced Numerical
14 Mar 2026
If ${\log _3}\,2\,,\,\,{\log _3}\,({2^x} - 5)\,,\,and\,\,{\log _3}\,\left( {{2^x} - {7 \over 2}} \right)$ are in arithmetic progression, determine the value of x.
1988 Q447 JEE Advanced MCQ
14 Mar 2026
Sum of the first n terms of the series ${1 \over 2} + {3 \over 4} + {7 \over 8} + {{15} \over {16}} + ............$ is equal to
A.
${2^n} - n - 1$
B.
$1 - {2^{ - n}}$
C.
$n + {2^{ - n}} - 1$
D.
${2^n} + 1$
1988 Q448 JEE Advanced MSQ
14 Mar 2026
If the first and the $(2n-1)$st terms of an A.P., a G.P. and an H.P. are equal and their $n$-th terms are $a,b$ and $c$ respectively, then
A.
$a = b = c$
B.
$a \ge b \ge c$
C.
$a + c = b$
D.
$ac - {b^2} = 0$
1988 Q449 JEE Advanced Numerical
14 Mar 2026
The sum of the first n terms of the series ${1^2} + {2.2^2} + {3^2} + {2.4^2} + {5^2} + {2.6^2} + .........$ is
$n\,\,{\left( {n + 1} \right)^2}/2,$ when $n$ is even. When $n$ is odd, the sum is .............
1986 Q450 JEE Advanced Numerical
14 Mar 2026
The solution of the equation $lo{g_7}$ $lo{g_5}$ $\left( {\sqrt {x + 5} + \sqrt x } \right) = 0$ is .............