Sequences & Series Questions - JEE

Find the sum: $1^2 + \dfrac{3^2}{1!} + \dfrac{5^2}{3!} + \dfrac{7^2}{5!} + \ldots$ to $\infty$.

Suppose p is the first of n (n \geq 1) arithmetic means between two positive numbers a and b. The value of p is

The first term of an infinite geometric series is 21. The second term and the sum of the series are both positive integers. Then which of the following is not the possible value of the second term?

For what value of n, $\dfrac{a^{n+1}+b^{n+1}}{a^n+b^n}$ is the geometric mean of a and b?

Let the A.P. be $a - 3d, a - d, a + d, a + 3d$. The sum of the terms = $48 - 4a = a = 12$. Given $\frac{(12 - 3d)^2}{(3 + d)^2} = \frac{23}{27}$. Find the sum.

If $(1 + 3 + 5 + \cdots + p) + (1 + 3 + 5 + \cdots + q) = (1 + 3 + 5 + \cdots + r)$ where each set of parentheses contains the sum of consecutive odd integers as shown, the smallest possible value of $p + q + r$ (where $p > 6$) is

Statement-1: For every natural number $n \geq 2$, $\dfrac{1}{\sqrt{1}} + \dfrac{1}{\sqrt{2}} + \cdots + \dfrac{1}{\sqrt{n}} > \sqrt{n}$.Statement-2: For every natural number $n \geq 2$, \(\sqrt{n(n+1)}

319. If $x > 0$ then the minimum value of $x^3 + \frac{1}{x^3}$ is

The sum of the first n terms of the series $1^2 + 2 \times 2^2 + 3^2 + 2 \times 4^2 + 5^2 + 2 \times 6^2 + \cdots$ is $\dfrac{n(n+1)^2}{2}$, when n is even. Find the sum when n is odd.

Let $S \subset (0, \pi)$ denote the set of values of x satisfying the equation $8^{1+|\cos x| + \cos^2 x + |\cos^3 x| + \cdots \text{ to } \infty} = 4^3$. Then $S =$

Find the sum $\dfrac{1\times2}{3!}+\dfrac{2\times2^2}{4!}+\dfrac{3\times2^3}{5!}+\cdots+\dfrac{20\times2^{20}}{22!}$.

37. Consider the ten numbers $ar, ar^2, ar^3, \ldots, ar^{10}$. If their sum is 18 and the sum of their reciprocals is 6, then the product of these ten numbers is

If a_1, a_2, a_3, \ldots are in AP, and a_1 + a_{30} = a_6 + a_{25} = a_{10} + a_{21} = S (say), with a_1 + a_6 + a_{10} + a_{21} + a_{25} + a_{30} = 120, then find \sum_{i=1}^{30} a_i.

If $S_p$ denotes the sum of the series $1 + r^p + r^{2p} + \cdots$ to ∞ and $s_p$ the sum of the series $1 - r^p + r^{2p} - r^{3p} + \cdots$ to ∞, \(|r|

If $1,\ \log_{\sqrt{3^{1-x}+2}},\ \log_3(4\cdot3^x - 1)$ are in AP, then $x$ equals

If $a + b + c = 20$, $2a + 2b + 2c = 50$, and $c^2 = 186$, find $c - a$.

The 5th and 8th terms of a geometric sequence of real numbers are 7! and 8!, respectively. If the sum to first $n$ terms of the G.P. is 2205, then $n$ equals ___.

Given $a_1, a_2, \ldots, a_{30}$ be an A.P. If $S = a_1 + a_2 + \cdots + a_{30}$, $T = \displaystyle\sum_{i=1}^{15} a_{(2i-1)}$, $a_5 = 27$ and $S - 2T = 75$, then find $a_{10}$.

If $a, b, c$ and $d$ are four positive real numbers such that $abcd = 1$, the minimum value of $(1 + a)(1 + b)(1 + c)(1 + d)$ is

If $ x + y + z = 12 $, find $ x^3 + y^3 + z^3 $ given $ x = 3 $, $ y = 4 $, $ z = 5 $ (using AM-GM with $ x/3 = y/4 = z/5 $).

If $S_n = \dfrac{1}{1^3} + \dfrac{1+2}{1^3+2^3} + \dfrac{1+2+3}{1^3+2^3+3^3} + \cdots + \dfrac{1+2+\cdots+n}{1^3+2^3+\cdots+n^3}$, then $S_n$ equals:

The sum of an infinite geometric series with positive terms is 3 and the sum of the cubes of its terms is $\dfrac{27}{19}$. Then the common ratio of this series is:

Find four numbers in G.P. whose sum is 85 and product is 4096.

Let $S = 1 + \frac{2}{3} + \frac{6}{3^2} + \frac{10}{3^3} + \frac{14}{3^4} + \cdots$. Find the value of $S$.

If $1^2 + 2^2 + 3^2 + \cdots + 2003^2 = (2003)(4007)(334)$ and $(1)(2003) + (2)(2002) + (3)(2001) + \cdots + (2003)(1) = (2003)(334)(x)$, then $x$ equals

If the sum of the series $\displaystyle\sum_{n=0}^{\infty} r^n$, \(|r|

Let $a, b, c \in \mathbb{R}$. If $f(x) = ax^2 + bx + c$ is such that $a + b + c = 3$ and $f(x+y) = f(x) + f(y) + xy$, $\forall\, x, y \in \mathbb{R}$, then $\displaystyle\sum_{n=1}^{10} f(n)$ is equal to

Find the sum to $n$ terms of the series,\[1 + \left(1 + \dfrac{1}{2} + \dfrac{1}{2^2}\right) + \left(1 + \dfrac{1}{2} + \dfrac{1}{2^2} + \dfrac{1}{2^3} + \dfrac{1}{2^4}\right) + \cdots\]

For Problems 25–27: Two arithmetic progressions have the same numbers. The ratio of the last term of the first progression to the first term of the second progression is equal to the ratio of the last term of the second progression to the first term of the first progression and is equal to 4. The ratio of the sum of the $n$ terms of the first progression to the sum of the $n$ terms of the second progression is equal to 2.The ratio of their $m$th term is

If the sides of a right-angled triangle are in A.P., then the sines of the acute angles are

The 1st, 2nd and 3rd terms of an arithmetic series are $a$, $b$ and $a^2$, where $a$ is negative. Then sum of an infinite geometric series whose first three terms are $a$, $a^2$ and $b$ respectively, is:

Let $\alpha$ and $\beta$ be two numbers where $\alpha < \beta$. The geometric mean of these numbers exceeds the smaller number $\alpha$ by 12 and the arithmetic mean of the same numbers is smaller by 24 than the larger number $\beta$, then the value of $|\beta - \alpha|$ is

Let $a_n$ be an infinite geometric sequence with a convergent and negative sum. The common ratio of the sequence is $r$ and the first term is $a_1$, then which one of the following is always true?

Let $x = 1.1$. $S = 1 + 2x + 3x^2 + \ldots + 10x^9$ and $xS = x + 2x^2 + \ldots + 9x^9 + 10x^{10}$. Subtracting we get $S(1-x) = (1 + x + x^2 + \ldots + x^9) - 10x^{10}$. Find $S$.

The value of $3\displaystyle\sum_{n=1}^{\infty} \left(\frac{1}{\pi}\sum_{k=1}^{\infty} \cot^{-1}\left(1+2\sqrt{\sum_{r=1}^{k} r^3}\right)\right)^n$ is less than:

The sum of the first ten terms of an AP is four times the sum of the first five terms, the ratio of the first term to the common difference is

Let the GP be \(a, ar, ar^2, \ldots (0

For Problems 13–15: Consider the sequence in the form of groups $(1), (2, 2), (3, 3, 3), (4, 4, 4, 4), (5, 5, 5, 5, 5), \ldots$The sum of the remaining terms in the group after 2000th term in which 2000th term lies is

If $a$, $\dfrac{1}{b}$, $c$ and $\dfrac{1}{p}$, $q$, $\dfrac{1}{r}$ form two arithmetic progressions of the same common difference, then $a$, $q$, $c$ are in A.P. if

If $S_n = \displaystyle\sum_{r=0}^{n} \dfrac{1}{{}^nC_r}$ and $t_n = \displaystyle\sum_{r=0}^{n} \dfrac{r}{{}^nC_r}$, then $\dfrac{t_n}{S_n}$ equals:

The common difference of the A.P. $a_1,a_2,\ldots,a_m$ is 13 more than the common difference of the A.P. $b_1,b_2,\ldots,b_n$. If $b_{31}=-277$, $b_{43}=-385$ and $a_{78}=327$, then $a_1$ is equal to

If $b_1, b_2, b_3$ ($b_1 > 0$) are three successive terms of a G.P. with common ratio $r$, the value of $r$ for which the inequality $b_3 > 4b_2 - 3b_1$ holds is given by

Consider the function g(x) defined as g(x) = [(x(2²⁰⁰⁸ − 1) − 1) = (x − 1)(x² − 1)(x⁴ − 1)⋯(x^{2^{2007}} − 1) − 1]. The value of g(2) equals ……

A man arranges to pay off a debt of ₹3600 by 40 annual instalments which are in AP. When 30 of the instalments are paid, he dies leaving one-third of the debt unpaid. The value of the 8th instalment is

162. If the first, fifth and last terms of an A.P. are $l, m, p$ respectively and the sum of A.P. is $\dfrac{(l+p)(4p+m-5l)}{k(m-l)}$, then $k$ is:

Let the first term $a$ and common ratio $r$ of a GP be positive integers. If sum of squares of first 3 terms is 33033, then sum of first 3 terms is

n arithmetic means are inserted between x and 2y and then between 2x and y. If the rth means in each case be equal, then find the ratio x/y.

If $S_n=4+11+21+34+50+\ldots$ to $n$ terms, then $\dfrac{1}{60}(S_{29}-S_9)$ is equal to

If the altitudes of triangle ABC are in harmonic progression, then the side length $b$ (which is CA) can be

Let $a_n$ be the $n$th term of $5+8+14+23+35+50+\ldots$ and $S_n=\sum_{k=1}^n a_k$. Then $S_{30}-a_{40}$ is equal to

Sequences & Series Questions (847)