Sequences and Series · Mathematics · JEE Advanced

MCQ (Single Correct Answer)

Let $${A_1}$$, $${G_1}$$, $${H_1}$$ denote the arithmetic, geometric and harmonic means, respectively, of two distinct positive numbers. For $$n \ge 2,\,Let\,{A_{n - 1}}\,\,and\,\,{H_{n - 1}}$$ have arithmetic, geometric and harminic means as $${A_n},{G_n}\,,{H_n}$$ repectively.

Which one of the following statements is correct ?

IIT-JEE 2007

Which one of the following statements is correct ?

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 $${T_r} = \,{V_{r + 1}} - \,{V_r} - 2\,\,\,and\,\,\,{Q_r} = \,{T_{r + 1}} - \,{T_r}\,for\,r = 1,2,...$$

Which one of the following is a correct statement?

IIT-JEE 2007

Which one of the following statements is correct ?

IIT-JEE 2007

$${T_r}$$ is always

IIT-JEE 2007

The sum $${V_1}$$+$${V_2}$$ +...+$${V_n}$$ is

IIT-JEE 2007

Let b_i > 1 for I = 1, 2, ......, 101. Suppose log_eb₁, log_eb₂, ......., log_eb₁₀₁ are in Arithmetic Progression (A.P.) with the common difference log_e2. Suppose a₁, a₂, ......, a₁₀₁ are in A.P. such that a₁ = b₁ and a₅₁ = b₅₁. If t = b₁ + b₂ + .... + b₅₁ and s = a₁ + a₂ + ..... + a₅₁, then

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 integer $$n$$ for which $${a_n} < 0$$ is

IIT-JEE 2012 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 2009 Paper 2 Offline

Suppose four distinct positive numbers $${a_1},\,{a_{2\,}},\,{a_3},\,{a_4}\,$$ are in G.P. Let $${b_1} = {a_1},{b_2} = {b_1} + {a_2},\,{b_3} = {b_2} + {a_{3\,\,}}\,\,\,and\,\,\,{b_4} = {b_3} + {a_4}$$.

STATEMENT-1: The numbers $${b_1},\,{b_{2\,}},\,{b_3},\,{b_4}\,$$ are neither in A.P. nor in G.P. and

STATEMENT-2 The numbers $${b_1},\,{b_{2\,}},\,{b_3},\,{b_4}\,$$ are in H.P.

IIT-JEE 2008 Paper 2 Offline

Which one of the following statements is correct?

IIT-JEE 2007 Paper 2 Offline

Which one of the following statements is correct?

IIT-JEE 2007 Paper 2 Offline

Which one of the following statements is correct?

IIT-JEE 2007 Paper 2 Offline

The sum V$$_1$$ + V$$_2$$ + ... + V$$_n$$ is

IIT-JEE 2007 Paper 1 Offline

T$$_r$$ is always

IIT-JEE 2007 Paper 1 Offline

Which one of the following is a correct statement?

IIT-JEE 2007 Paper 1 Offline

In the quadratic equation $$\,\,a{x^2} + bx + c = 0,$$ $$\Delta $$ $$ = {b^2} - 4ac$$ and $$\alpha + \beta ,\,{\alpha ^2} + {\beta ^2},\,{\alpha ^3} + {\beta ^3},$$ are in G.P. where $$\alpha ,\beta $$ are the root of $$\,\,a{x^2} + bx + c = 0,$$ then

IIT-JEE 2005 Screening

If total number of runs scored in $$n$$ matches is $$\left(\frac{n+1}{4}\right)\left(2^{n+1}-n-2\right)$$ where $$n > 1$$, and the runs scored in the $$k^{\text {th }}$$ match are given by $$k .2^{n+1-k}$$, where $$1 \leq k \leq n$$. Find, $$n$$.

IIT-JEE 2005 Mains

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 $$a + b + c = {3 \over 2},$$ then the value of $$a$$ is

IIT-JEE 2002 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 $$\alpha ,\beta ,\gamma ,\delta $$ are in G.P., then the integral values of $$p$$ and $$q$$ respectively, are

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 $$n$$ equals

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 3/4, then

IIT-JEE 2000 Screening

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 $${a_{10}} = {h_{10}} = 3,$$ then $${a_4}{h_7}$$ is

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 1999

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
$${T_m} = {1 \over n}$$ and $${T_n} = {1 \over m},$$ then $${T_n} = {1 \over m},$$ equals

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 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 equal to

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 common root if $${d \over a},\,{e \over b},{f \over c}$$ are in ________.

IIT-JEE 1985

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

IIT-JEE 1983

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 $${x^{y - z}}\,{y^{z - x}}\,{z^{x - y}}$$ is equal to :

IIT-JEE 1982

Numerical

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 . Consider the sum $S=77+757+7557+\cdots+7 \overbrace{5 \cdots 5}^{98}7$. If $S=\frac{7 \overbrace{5 \cdots 5}^{99}7+m}{n}$, where $m$ and $n$ are natural numbers less than 3000 , then the value of $m+n$ is

JEE Advanced 2023 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 $$w_{1}, w_{2}, \ldots, w_{100}$$ be consecutive terms of another arithmetic progression with common difference $$d_{2}$$, where $$d_{1} d_{2}=10$$. For each $$i=1,2, \ldots, 100$$, let $R_{i}$ be a rectangle with length $$l_{i}$$, width $$w_{i}$$ and area $A_{i}$. If $$A_{51}-A_{50}=1000$$, then the value of $$A_{100}-A_{90}$$ is __________.

JEE Advanced 2022 Paper 1 Online

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 $${{y_1} + {y_2} + {y_3}}$$ = 9. Let M be the maximum possible value of $$({\log _3}{x_1} + {\log _3}{x_2} + {\log _3}{x_3})$$, where $${x_1},{x_2},{x_3}$$ are positive real numbers for which $${{x_1} + {x_2} + {x_3}}$$ = 9. Then the value of $${\log _2}({m^3}) + {\log _3}({M^2})$$ is ...........

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 of positive integers in geometric progression with common ratio 2. If a₁ = b₁ = c, then the number of all possible values of c, for which the equality 2(a₁ + a₂ + ... + a_n) = b₁ + b₂ + ... + b_n holds for some positive integer n, is ...........

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 difference d > 0. If $$AP(1;3) \cap AP(2;5) \cap AP(3;7)$$ = AP(a ; d), then a + d equals ..............

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 consisting of the first 2018 terms of the arithmetic progression 9, 16, 23, .... . Then, the number of elements in the set X $$ \cup $$ Y is .........

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 of its smallest side?

JEE Advanced 2017 Paper 1 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 first eleven terms is 6 : 11 and the seventh term lies in between 130 and 140, then the common difference of this A.P. is

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 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 the arithmetic mean of a, b, c is b + 2, then the value of $${{{a^2} + a - 14} \over {a + 1}}$$ is

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 the sum of the numbers on the remaining cards is $$1224.$$ If the smaller of the numbers on the removed cards is $$k,$$ then $$k-20=$$

JEE Advanced 2013 Paper 1 Offline

Let $${{a_1}}$$, $${{a_2}}$$, $${{a_3}}$$........ $${{a_{100}}}$$ be an arithmetic progression with $${{a_1}}$$ = 3 and $${S_p} = \sum\limits_{i = 1}^p {{a_i},1 \le } \,p\, \le 100$$. For any integer n with $$1\,\, \le \,n\, \le 20$$, let m = 5n. If $${{{S_m}} \over {{S_n}}}$$ does not depend on n, then $${a_{2\,}}$$ is

IIT-JEE 2011 Paper 1 Offline

Let $${S_k}$$= 1, 2,....., 100, denote the sum of the infinite geometric series whose first term is $$\,{{k - 1} \over {k\,!}}$$ and the common ratio is $${1 \over k}$$. Then the value of $${{{{100}^2}} \over {100!}}\,\, + \,\,\sum\limits_{k = 1}^{100} {\left| {({k^2} - 3k + 1)\,\,{S_k}} \right|\,\,} $$ is

IIT-JEE 2010 Paper 1 Offline

Let $${a_1},\,{a_{2\,}},\,{a_3}$$......,$${a_{11}}$$ be real numbers satisfying $${a_1} = 15,27 - 2{a_2} > 0\,\,and\,\,{a_k} = 2{a_{k - 1}} - {a_{k - 2}}\,\,for\,k = 3,4,........11$$. if $$\,\,\,{{a_1^2 + a_2^2 + .... + a_{11}^2} \over {11}} = 90$$, then the value of $${{{a_1} + {a_2} + .... + {a_{11}}} \over {11}}$$ is equal to :

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 arithmetic progression, determine the value of x.

IIT-JEE 1990

MCQ (More than One Correct Answer)

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 that $$T_{1}=3$$ and $$T_{n+1}-T_{n}=a_{n}$$ for $$n \geq 1$$. Then, which of the following is/are TRUE ?

JEE Advanced 2022 Paper 1 Online

For any positive integer n, let S_n : (0, $$\infty$$) $$\to$$ R be defined by $${S_n}(x) = \sum\nolimits_{k = 1}^n {{{\cot }^{ - 1}}\left( {{{1 + k(k + 1){x^2}} \over x}} \right)} $$, where for any x $$\in$$ R, $${\cot ^{ - 1}}(x) \in (0,\pi )$$ and $${\tan ^{ - 1}}(x) \in \left( { - {\pi \over 2},{\pi \over 2}} \right)$$. Then which of the following statements is (are) TRUE?

JEE Advanced 2021 Paper 1 Online

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)

JEE Advanced 2013 Paper 1 Offline

Let $${S_n} = \sum\limits_{k = 1}^n {{n \over {{n^2} + kn + {k^2}}}} $$ and $${T_n} = \sum\limits_{k = 0}^{n - 1} {{n \over {{n^2} + kn + {k^2}}}} $$ for $$n$$ $$=1, 2, 3, ............$$ Then,

IIT-JEE 2008 Paper 1 Offline

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

IIT-JEE 1999

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

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 $$a,b$$ and $$c$$ respectively, then

IIT-JEE 1988

Subjective

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 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 in geometric progression and a, $${{H_1},{H_2}}$$, b are in harmonic progression, show that $$\,{{{G_1},{G_2}} \over {{H_1},{H_2}}} = {{{A_1} + {A_2}} \over {{H_1} + {H_2}}} = {{(2a + b)\,(a + 2b)} \over {9ab}}$$.

IIT-JEE 2002

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 respectively, the arithmetic mean , geometric mean, and harmonic mean of $${a_1}$$,$${a_2}$$......,$${a_n}$$. Find an expression for the geometric mean of $${G_1}$$,$${G_2}$$,.....,$${G_n}$$ in terms of $${A_1}$$,$${A_2}$$,.....,$${A_n}$$,$${H_n}$$,$${H_1}$$,$${H_2}$$,........,$${H_n}$$.

IIT-JEE 2001

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 it. Prove that the resulting sum is the square of an integer.

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
6u + 9v = 4

then show that the roots of the equation $$\left( {{1 \over u} + {1 \over v} + {1 \over w}} \right){x^2}$$
$$ + [{(b - c)^2} + {(c - a)^2} + {(d - b)^2}]x + u + v + w = 0$$ and $$20{x^2} + 10{(a - d)^2}x - 9 = 0$$ are reciprocals of each other.

IIT-JEE 1999

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.

IIT-JEE 1996

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 $$\,{\left( {{{n + 1} \over {n - 1}}} \right)^2}\,p$$.

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 $${S_1}^2 + {S_2}^2 + {S_3}^2 + ....... + {S^2}_{2n - 1}$$

IIT-JEE 1991

Find the sum of the series : $$$\sum\limits_{r = 0}^n {{{\left( { - 1} \right)}^r}\,{}^n{C_r}\left[ {{1 \over {{2^r}}} + {{{3^r}} \over {{2^{2r}}}} + {{{7^r}} \over {{2^{3r}}}} + {{{{15}^r}} \over {{2^{4r}}}}..........up\,\,to\,\,m\,\,terms} \right]} $$$

IIT-JEE 1985

If $$a > 0,\,b > 0$$ and $$\,c > 0,$$ prove that $$\,c > 0,$$ prove that $$\left( {a + b + c} \right)\left( {{1 \over a} + {1 \over b} + {1 \over c}} \right) \ge 9$$

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