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JEE Advanced

Sequences and Series

Mathematics

Previous Years Questions

Numerical

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Let a1, a2, a3, .... be a sequence of positive integers in arithmetic progression with common difference 2. Also, let b1...
JEE Advanced 2020 Paper 1 Offline
Let m be the minimum possible value of $${\log _3}({3^{{y_1}}} + {3^{{y_2}}} + {3^{{y_3}}})$$, where $${y_1},{y_2},{y_3}...
JEE Advanced 2020 Paper 1 Offline
Let AP(a; d) denote the set of all the terms of an infinite arithmetic progression with first term a and common differen...
JEE Advanced 2019 Paper 1 Offline
Let X be the set consisting of the first 2018 terms of the arithmetic progression 1, 6, 11, ...., and Y be the set consi...
JEE Advanced 2018 Paper 1 Offline
The sides of a right angled triangle are in arithmetic progression. If the triangle has area 24, then what is the length...
JEE Advanced 2017 Paper 1 Offline
The coefficient of $${x^9}$$ in the expansion of (1 + x) (1 + $${x^2)}$$ (1 + $${x^3}$$) ....$$(1 + {x^{100}})$$ is
JEE Advanced 2015 Paper 2 Offline
Suppose that all the terms of an arithmetic progression (A.P) are natural numbers. If the ratio of the sum of the first ...
JEE Advanced 2015 Paper 2 Offline
Let a, b, c be positive integers such that $${b \over a}$$ is an integer. If a, b, c are in geometric progression and th...
JEE Advanced 2014 Paper 1 Offline
A pack contains $$n$$ cards numbered from $$1$$ to $$n.$$ Two consecutive numbered cards are removed from the pack and t...
JEE Advanced 2013 Paper 1 Offline
A pack contains n cards numbered from 1 to n. Two consecutive numbered cards are removed from the pack and the sum of th...
JEE Advanced 2013 Paper 1 Offline
Let $${{a_1}}$$, $${{a_2}}$$, $${{a_3}}$$........ $${{a_{100}}}$$ be an arithmetic progression with $${{a_1}}$$ = 3 and ...
IIT-JEE 2011 Paper 1 Offline
Let $${S_k}$$= 1, 2,....., 100, denote the sum of the infinite geometric series whose first term is $$\,{{k - 1} \over {...
IIT-JEE 2010 Paper 1 Offline
Let $${a_1},\,{a_{2\,}},\,{a_3}$$......,$${a_{11}}$$ be real numbers satisfying $${a_1} = 15,27 - 2{a_2} > 0\,\,and\,...
IIT-JEE 2010 Paper 2 Offline
If $${\log _3}\,2\,,\,\,{\log _3}\,({2^x} - 5)\,,\,and\,\,{\log _3}\,\left( {{2^x} - {7 \over 2}} \right)$$ are in arith...
IIT-JEE 1990

MCQ (Single Correct Answer)

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Let bi > 1 for I = 1, 2, ......, 101. Suppose logeb1, logeb2, ......., logeb101 are in Arithmetic Progression (A.P.) wit...
JEE Advanced 2016 Paper 2 Offline
Let $${a_1},{a_2},{a_3},.....$$ be in harmonic progression with $${a_1} = 5$$ and $${a_{20}} = 25.$$ The least positive ...
IIT-JEE 2012 Paper 2 Offline
In the sum of first $$n$$ terms of an A.P. is $$c{n^\angle }$$, then the sum of squares of these $$n$$ terms is
IIT-JEE 2009
Suppose four distinct positive numbers $${a_1},\,{a_{2\,}},\,{a_3},\,{a_4}\,$$ are in G.P. Let $${b_1} = {a_1},{b_2} = {...
IIT-JEE 2008
Let $${A_1}$$, $${G_1}$$, $${H_1}$$ denote the arithmetic, geometric and harmonic means, respectively, of two distinct p...
IIT-JEE 2007
Let $$\,{V_r}$$ denote the sum of first r terms of an arithmetic progression (A.P.) whose first term is r and the common...
IIT-JEE 2007
Let $${A_1}$$, $${G_1}$$, $${H_1}$$ denote the arithmetic, geometric and harmonic means, respectively, of two distinct p...
IIT-JEE 2007
Let $${A_1}$$, $${G_1}$$, $${H_1}$$ denote the arithmetic, geometric and harmonic means, respectively, of two distinct p...
IIT-JEE 2007
Let $$\,{V_r}$$ denote the sum of first r terms of an arithmetic progression (A.P.) whose first term is r and the common...
IIT-JEE 2007
Let $$\,{V_r}$$ denote the sum of first r terms of an arithmetic progression (A.P.) whose first term is r and the common...
IIT-JEE 2007
In the quadratic equation $$\,\,a{x^2} + bx + c = 0,$$ $$\Delta $$ $$ = {b^2} - 4ac$$ and $$\alpha + \beta ,\,{\alpha ^...
IIT-JEE 2005 Screening
An infinite G.P. has first term '$$x$$' and sum '$$5$$', then $$x$$ belongs to
IIT-JEE 2004 Screening
Suppose $$a, b, c$$ are in A.P. and $${a^2},{b^2},{c^2}$$ are in G.P. If $$a < b < c$$ and $$a + b + c = {3 \over ...
IIT-JEE 2002 Screening
If the sum of the first $$2n$$ terms of the A.P.$$2,5,8,......,$$ is equal to the sum of the first $$n$$ terms of the A....
IIT-JEE 2001 Screening
Let the positive numbers $$a,b,c,d$$ be in A.P. Then $$abc,$$ $$abd,$$ $$acd,$$ $$bcd,$$ are
IIT-JEE 2001 Screening
Let $$\alpha $$, $$\beta $$ be the roots of $${x^2} - x + p = 0$$ and $$\gamma ,\delta $$ be the roots of $${x^2} - 4x +...
IIT-JEE 2001 Screening
Consider an infinite geometric series with first term a and common ratio $$r$$. If its sum is 4 and the second term is 3...
IIT-JEE 2000 Screening
The harmonic mean of the roots of the equation $$\left( {5 + \sqrt 2 } \right){x^2} - \left( {4 + \sqrt 5 } \right)x + 8...
IIT-JEE 1999
Let $${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...
IIT-JEE 1999
If $$x > 1,y > 1,z > 1$$ are in G.P., then $${1 \over {1 + In\,x}},{1 \over {1 + In\,y}},{1 \over {1 + In\,z}}$...
IIT-JEE 1998
Let $${T_r}$$ be the $${r^{th}}$$ term of an A.P., for $$r=1, 2, 3, ....$$ If for some positive integers $$m$$, $$n$$ we...
IIT-JEE 1998
Let $$n$$ be an odd integer. If $$\sin n\theta = \sum\limits_{r = 0}^n {{b_r}{{\sin }^r}\theta ,} $$ for every value of...
IIT-JEE 1998
If $$In\left( {a + c} \right),In\left( {a - c} \right),In\left( {a - 2b + c} \right)$$ are in A.P., then
IIT-JEE 1994
The number $${\log _2}\,7$$ is
IIT-JEE 1990
Sum of the first n terms of the series $${1 \over 2} + {3 \over 4} + {7 \over 8} + {{15} \over {16}} + ............$$ is...
IIT-JEE 1988
If $$a,\,b,\,c$$ are in GP., then the equations $$\,\,\alpha {x^2} + 2bx + c = 0$$ and $$d{x^2} + 2ex + f = 0$$ have a c...
IIT-JEE 1985
The rational number, which equals the number $$2\overline {357} $$ with recurring decimal is
IIT-JEE 1983
If $$x,\,y$$ and $$z$$ are $$pth$$, $$qth$$ and $$rth$$ terms respectively of an A.P. and also of a G.P., then $${x^{y -...
IIT-JEE 1982
The third term of a geometric progression is 4. The product of the first five terms is
IIT-JEE 1982

MCQ (More than One Correct Answer)

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Let $${S_n} = {\sum\limits_{k = 1}^{4n} {\left( { - 1} \right)} ^{{{k\left( {k + 1} \right)} \over 2}}}{k^2}.$$ Then $${...
JEE Advanced 2013 Paper 1 Offline
Let $${S_n} = \sum\limits_{k = 1}^n {{n \over {{n^2} + kn + {k^2}}}} $$ and $${T_n} = \sum\limits_{k = 0}^{n - 1} {{n \o...
IIT-JEE 2008
A straight line through the vertex p of a triangle PQR intersects the side QR at the point S and the circumcircle of the...
IIT-JEE 2008
For a positive integer $$n$$, let $$a\left( n \right) = 1 + {1 \over 2} + {1 \over 3} + {1 \over 4} + .....\,{1 \over {...
IIT-JEE 1999
For $$0 < \phi < \pi /2,$$ if $$x = $$$$\sum\limits_{n = 0}^\infty {{{\cos }^{2n}}\phi ,y = \sum\limits_{n = 0}...
IIT-JEE 1993
If the first and the $$(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$$ ...
IIT-JEE 1988

Subjective

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If $${a_n} = {3 \over 4} - {\left( {{3 \over 4}} \right)^2} + {\left( {{3 \over 4}} \right)^3} + ....{( - 1)^{n - 1}}{\l...
IIT-JEE 2006
If a, b, c are in A.P., $${a^2}$$, $${b^2}$$, $${c^2}$$ are in H.P., then prove that either a = b = c or a, b, $${ - {c ...
IIT-JEE 2003
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...
IIT-JEE 2002
Let $${a_1}$$, $${a_2}$$,.....,$${a_n}$$ be positive real numbers in geometric progression. For each n, let $${A_n}$$, $...
IIT-JEE 2001
The fourth power of the common difference of an arithmatic progression with integer entries is added to the product of a...
IIT-JEE 2000
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 ...
IIT-JEE 1999
The real numbers $${x_1}$$, $${x_2}$$, $${x_3}$$ satisfying the equation $${x^3} - {x^2} + \beta x + \gamma = 0$$ are i...
IIT-JEE 1996
If $${S_1}$$, $${S_2}$$, $${S_3}$$,.............,$${S_n}$$ are the sums of infinite geometric series whose first terms a...
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 nu...
IIT-JEE 1991
Solve for x the following equation: $${\log _{(2x + 3)}}(6{x^2} + 23x + 21) = 4 - {\log _{(3x + 7)}}(4{x^2} + 12x + 9)\...
IIT-JEE 1987
Find the sum of the series : $$$\sum\limits_{r = 0}^n {{{\left( { - 1} \right)}^r}\,{}^n{C_r}\left[ {{1 \over {{2^r}}} +...
IIT-JEE 1985
If $$n$$ is a natural number such that $$n = {p_1}{}^{{\alpha _1}}{p_2}{}^{{\alpha _2}}.{p_3}{}^{{\alpha _3}}........{p...
IIT-JEE 1984
If $$a > 0,\,b > 0$$ and $$\,c > 0,$$ prove that $$\,c > 0,$$ prove that $$\left( {a + b + c} \right)\left( ...
IIT-JEE 1984
Find three numbers $$a,b,c$$ between $$2$$ and $$18$$ such that (i) their sum is $$25$$ (ii) the numbers $$2,$$ $$a, b$...
IIT-JEE 1983
Does there exist a geometric progression containing $$27, 8$$ and $$12$$ as three of its terms? If it exits, how many su...
IIT-JEE 1982
The interior angles of a polygon are in arithmetic progression. The smallest angle is $${120^ \circ }$$, and the common ...
IIT-JEE 1980
The harmonic mean of two numbers is 4.Their arithmetic mean $$A$$ and the geometric mean $$G$$ satisfy the relation. $$2...
IIT-JEE 1979

Fill in the Blanks

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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...
IIT-JEE 1997
For any odd integer $$n$$ $$ \ge 1,\,\,{n^3} - {\left( {n - 1} \right)^3} + .... + {\left( { - 1} \right)^{n - 1}}\,{1^3...
IIT-JEE 1996
Let the harmonic mean and geometric mean of two positive numbers be the ratio 4 : 5. Then the two number are in the rati...
IIT-JEE 1992
The sum of the first n terms of the series $${1^2} + {2.2^2} + {3^2} + {2.4^2} + {5^2} + {2.6^2} + .........$$ is $$n\,...
IIT-JEE 1988
The solution of the equation $$lo{g_7}$$ $$lo{g_5}$$ $$\left( {\sqrt {x + 5} + \sqrt x } \right) = 0$$ is ................
IIT-JEE 1986
The sum of integers from 1 to 100 that are divisible by 2 or 5 is ............
IIT-JEE 1984

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