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Sequences and Series

426 Questions
All Exams JEE Mains JEE Advanced TS-EAMCET AP-EAPCET
1997 JEE Advanced Numerical
IIT-JEE 1997
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 JEE Advanced Numerical
IIT-JEE 1996
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 JEE Advanced Numerical
IIT-JEE 1996
For any odd integer $n$ $ \ge 1,\,\,{n^3} - {\left( {n - 1} \right)^3} + .... + {\left( { - 1} \right)^{n - 1}}\,{1^3} = ........$
1994 JEE Advanced MCQ
IIT-JEE 1994
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 JEE Advanced MSQ
IIT-JEE 1993
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 JEE Advanced Numerical
IIT-JEE 1992
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 JEE Advanced Numerical
IIT-JEE 1991
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 JEE Advanced Numerical
IIT-JEE 1991
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 JEE Advanced MCQ
IIT-JEE 1990
The number ${\log _2}\,7$ is
A.
an integer
B.
a rational number
C.
an irrational number
D.
a prime number
1990 JEE Advanced Numerical
IIT-JEE 1990
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 JEE Advanced MCQ
IIT-JEE 1988
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 JEE Advanced MSQ
IIT-JEE 1988
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 JEE Advanced Numerical
IIT-JEE 1988
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 JEE Advanced Numerical
IIT-JEE 1986
The solution of the equation $lo{g_7}$ $lo{g_5}$ $\left( {\sqrt {x + 5} + \sqrt x } \right) = 0$ is .............
1985 JEE Advanced MCQ
IIT-JEE 1985
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 JEE Advanced Numerical
IIT-JEE 1985
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 JEE Advanced Numerical
IIT-JEE 1984
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 JEE Advanced Numerical
IIT-JEE 1984
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 JEE Advanced Numerical
IIT-JEE 1984
The sum of integers from 1 to 100 that are divisible by 2 or 5 is ............
1983 JEE Advanced MCQ
IIT-JEE 1983
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 JEE Advanced Numerical
IIT-JEE 1983
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 JEE Advanced MCQ
IIT-JEE 1982
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 JEE Advanced MCQ
IIT-JEE 1982
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 JEE Advanced Numerical
IIT-JEE 1982
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 JEE Advanced Numerical
IIT-JEE 1980
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 JEE Advanced Numerical
IIT-JEE 1979
The harmonic mean of two numbers is 4.Their arithmetic mean $A$ and the geometric mean $G$ satisfy the relation. $2A + {G^2} = 27$