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Sequences and Series

79 Questions Start JEE Advanced Test
All Exams JEE Mains JEE Advanced TS-EAMCET AP-EAPCET BITSAT
1998 Q51 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 Q52 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 Q53 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 Q54 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 Q55 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 Q56 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 Q57 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 Q58 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 Q59 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 Q60 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 Q61 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 Q62 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 Q63 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 Q64 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 Q65 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 Q66 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 Q67 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 .............
1985 Q68 JEE Advanced MCQ
14 Mar 2026
If $a,\,b,\,c$ are in GP., then the equations $\,\,\alpha {x^2} + 2bx + c = 0$ and $d{x^2} + 2ex + f = 0$ have a common root if ${d \over a},\,{e \over b},{f \over c}$ are in ________.
A.
A.P.
B.
GP.
C.
H.P.
D.
none of these
1985 Q69 JEE Advanced Numerical
14 Mar 2026
Find the sum of the series : $$\sum\limits_{r = 0}^n {{{\left( { - 1} \right)}^r}\,{}^n{C_r}\left[ {{1 \over {{2^r}}} + {{{3^r}} \over {{2^{2r}}}} + {{{7^r}} \over {{2^{3r}}}} + {{{{15}^r}} \over {{2^{4r}}}}..........up\,\,to\,\,m\,\,terms} \right]} $$
1984 Q70 JEE Advanced Numerical
14 Mar 2026
If $n$ is a natural number such that
$n = {p_1}{}^{{\alpha _1}}{p_2}{}^{{\alpha _2}}.{p_3}{}^{{\alpha _3}}........{p_k}{}^{{\alpha _k}}$ and ${p_1},{p_2},\,\,......,\,{p_k}$ are distinct primes, then show that $In$ $n \ge k$ $in$ 2
1984 Q71 JEE Advanced Numerical
14 Mar 2026
If $a > 0,\,b > 0$ and $\,c > 0,$ prove that $\,c > 0,$ prove that $\left( {a + b + c} \right)\left( {{1 \over a} + {1 \over b} + {1 \over c}} \right) \ge 9$
1984 Q72 JEE Advanced Numerical
14 Mar 2026
The sum of integers from 1 to 100 that are divisible by 2 or 5 is ............
1983 Q73 JEE Advanced MCQ
14 Mar 2026
The rational number, which equals the number $2\overline {357} $ with recurring decimal is
A.
${{2355} \over {1001}}$
B.
${{2379} \over {997}}$
C.
${{2355} \over {999}}$
D.
none of these
1983 Q74 JEE Advanced Numerical
14 Mar 2026
Find three numbers $a,b,c$ between $2$ and $18$ such that
(i) their sum is $25$
(ii) the numbers $2,$ $a, b$ are consecutive terms of an A.P. and
(iii) the numbers $b,c,18$ are consecutive terms of a G.P.
1982 Q75 JEE Advanced MCQ
14 Mar 2026
The third term of a geometric progression is 4. The product of the first five terms is
A.
43
B.
45
C.
44
D.
none of these
1982 Q76 JEE Advanced MCQ
14 Mar 2026
If $x,\,y$ and $z$ are $pth$, $qth$ and $rth$ terms respectively of an A.P. and also of a G.P., then ${x^{y - z}}\,{y^{z - x}}\,{z^{x - y}}$ is equal to :
A.
$xyz$
B.
$0$
C.
$1$
D.
None of these
1982 Q77 JEE Advanced Numerical
14 Mar 2026
Does there exist a geometric progression containing $27, 8$ and $12$ as three of its terms? If it exits, how many such progressions are possible ?
1980 Q78 JEE Advanced Numerical
14 Mar 2026
The interior angles of a polygon are in arithmetic progression. The smallest angle is ${120^ \circ }$, and the common difference is ${5^ \circ }$, Find the number of sides of the polygon.
1979 Q79 JEE Advanced Numerical
14 Mar 2026
The harmonic mean of two numbers is 4.Their arithmetic mean $A$ and the geometric mean $G$ satisfy the relation. $2A + {G^2} = 27$