# Sequences and Series · Mathematics · JEE Advanced

Start Practice## Numerical

JEE Advanced 2023 Paper 1 Online

Let $7 \overbrace{5 \cdots 5}^r 7$ denote the $(r+2)$ digit number where the first and the last digits are 7 and the remaining $r$ digits are 5 . Cons...

JEE Advanced 2022 Paper 1 Online

Let $$l_{1}, l_{2}, \ldots, l_{100}$$ be consecutive terms of an arithmetic progression with common difference $$d_{1}$$, and let $$w_{1}, w_{2}, \ldo...

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 $${y_1},{y_2},{y_3}$$ are real numbers for which ...

JEE Advanced 2020 Paper 1 Offline

Let a1, a2, a3, .... be a sequence of positive integers in arithmetic progression with common difference 2. Also, let b1, b2, b3, .... be a sequence o...

JEE Advanced 2019 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 difference d > 0. If $$AP(1;3) \cap...

JEE Advanced 2018 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 consisting of the first 2018 terms ...

JEE Advanced 2017 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 of its smallest side?

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 seven terms to the sum of the ...

JEE Advanced 2015 Paper 2 Offline

The coefficient of $${x^9}$$ in the expansion of (1 + x) (1 + $${x^2)}$$ (1 + $${x^3}$$) ....$$(1 + {x^{100}})$$ is

JEE Advanced 2014 Paper 1 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 the arithmetic mean of a, b, c i...

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 the numbers on the remaining car...

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 the numbers on the r...

IIT-JEE 2011 Paper 1 Offline

Let $${{a_1}}$$, $${{a_2}}$$, $${{a_3}}$$........ $${{a_{100}}}$$ be an arithmetic progression with $${{a_1}}$$ = 3 and $${S_p} = \sum\limits_{i = 1}^...

IIT-JEE 2010 Paper 1 Offline

Let $${S_k}$$= 1, 2,....., 100, denote the sum of the infinite geometric series whose first term is $$\,{{k - 1} \over {k\,!}}$$ and the common ratio ...

IIT-JEE 2010 Paper 2 Offline

Let $${a_1},\,{a_{2\,}},\,{a_3}$$......,$${a_{11}}$$ be real numbers satisfying $${a_1} = 15,27 - 2{a_2} > 0\,\,and\,\,{a_k} = 2{a_{k - 1}} - {a_{k...

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 t...

## MCQ (More than One Correct Answer)

JEE Advanced 2022 Paper 1 Online

Let $$a_{1}, a_{2}, a_{3}, \ldots$$ be an arithmetic progression with $$a_{1}=7$$ and common difference 8. Let $$T_{1}, T_{2}, T_{3}, \ldots$$ be such...

JEE Advanced 2013 Paper 1 Offline

Let $${S_n} = {\sum\limits_{k = 1}^{4n} {\left( { - 1} \right)} ^{{{k\left( {k + 1} \right)} \over 2}}}{k^2}.$$ Then $${S_n}$$can take value(s)

IIT-JEE 2008 Paper 1 Offline

Let $${S_n} = \sum\limits_{k = 1}^n {{n \over {{n^2} + kn + {k^2}}}} $$ and $${T_n} = \sum\limits_{k = 0}^{n - 1} {{n \over {{n^2} + kn + {k^2}}}} $$ ...

IIT-JEE 1999

For a positive integer $$n$$, let
$$a\left( n \right) = 1 + {1 \over 2} + {1 \over 3} + {1 \over 4} + .....\,{1 \over {\left( {{2^n}} \right) - 1}}$$...

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 ,...

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

## MCQ (Single Correct Answer)

JEE Advanced 2016 Paper 2 Offline

Let bi > 1 for I = 1, 2, ......, 101. Suppose logeb1, logeb2, ......., logeb101 are in Arithmetic Progression (A.P.) with the common difference loge2....

IIT-JEE 2012 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 integer $$n$$ for which $${a_n...

IIT-JEE 2009 Paper 2 Offline

If the sum of first $$n$$ terms of an A.P. is $$c{n^2}$$, then the sum of squares of these $$n$$ terms is

IIT-JEE 2008 Paper 2 Offline

Suppose four distinct positive numbers $${a_1},\,{a_{2\,}},\,{a_3},\,{a_4}\,$$ are in G.P. Let $${b_1} = {a_1},{b_2} = {b_1} + {a_2},\,{b_3} = {b_2} +...

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 difference is (2r-1). Let $${...

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 difference is (2r-1). Let $${...

IIT-JEE 2007

Let $${A_1}$$, $${G_1}$$, $${H_1}$$ denote the arithmetic, geometric and harmonic means, respectively, of two distinct positive numbers. For $$n \ge 2...

IIT-JEE 2007

Let $${A_1}$$, $${G_1}$$, $${H_1}$$ denote the arithmetic, geometric and harmonic means, respectively, of two distinct positive numbers. For $$n \ge 2...

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 difference is (2r-1). Let $${...

IIT-JEE 2007

Let $${A_1}$$, $${G_1}$$, $${H_1}$$ denote the arithmetic, geometric and harmonic means, respectively, of two distinct positive numbers. For $$n \ge 2...

IIT-JEE 2005 Screening

In the quadratic equation $$\,\,a{x^2} + bx + c = 0,$$ $$\Delta $$ $$ = {b^2} - 4ac$$ and $$\alpha + \beta ,\,{\alpha ^2} + {\beta ^2},\,{\alpha ^3} ...

IIT-JEE 2004 Screening

An infinite G.P. has first term '$$x$$' and sum '$$5$$', then $$x$$ belongs to

IIT-JEE 2002 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 $$a + b + c = {3 \over 2},$$ then the value of $$a$$ ...

IIT-JEE 2001 Screening

Let $$\alpha $$, $$\beta $$ be the roots of $${x^2} - x + p = 0$$ and $$\gamma ,\delta $$ be the roots of $${x^2} - 4x + q = 0.$$ If $$\alpha ,\beta ,...

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

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...

IIT-JEE 2000 Screening

Consider an infinite geometric series with first term a and common ratio $$r$$. If its sum is 4 and the second term is 3/4, then

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 $${a_1} = {h_1} = 2$$ and $${a_{10}} = {h_{10}} =...

IIT-JEE 1999

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

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 $$\theta ,$$ then

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 have
$${T_m} = {1 \over n}$$...

IIT-JEE 1998

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

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

IIT-JEE 1990

The number $${\log _2}\,7$$ is

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

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},\,...

IIT-JEE 1983

The rational number, which equals the number $$2\overline {357} $$ with recurring decimal is

IIT-JEE 1982

The third term of a geometric progression is 4. The product of the first five terms is

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}}...

## Subjective

IIT-JEE 2006

If $${a_n} = {3 \over 4} - {\left( {{3 \over 4}} \right)^2} + {\left( {{3 \over 4}} \right)^3} + ....{( - 1)^{n - 1}}{\left( {{3 \over 4}} \right)^n}\...

IIT-JEE 2003

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.

IIT-JEE 2002

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...

IIT-JEE 2001

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 respect...

IIT-JEE 2000

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 i...

IIT-JEE 1999

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 t...

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 wh...

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 li...

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, .....................

IIT-JEE 1987

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 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}}}} + ...

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} + {...

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...

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 ...

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 ?...

IIT-JEE 1980

The interior angles of a polygon are in arithmetic progression. The smallest angle is $${120^ \circ }$$, and the common difference is $${5^ \circ }$$,...

IIT-JEE 1979

The harmonic mean of two numbers is 4.Their arithmetic mean $$A$$ and the geometric mean $$G$$ satisfy the relation. $$2A + {G^2} = 27$$

## Fill in the Blanks

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 $...

IIT-JEE 1996

For any odd integer $$n$$ $$ \ge 1,\,\,{n^3} - {\left( {n - 1} \right)^3} + .... + {\left( { - 1} \right)^{n - 1}}\,{1^3} = ........$$

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 .........

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...

IIT-JEE 1986

The solution of the equation $$lo{g_7}$$ $$lo{g_5}$$ $$\left( {\sqrt {x + 5} + \sqrt x } \right) = 0$$ is .............

IIT-JEE 1984

The sum of integers from 1 to 100 that are divisible by 2 or 5 is ............