Sequences and Series · Mathematics · JEE Advanced

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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 ............
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