Sequences & Series Questions - JEE

Let $A=\{n\in[100,700]\cap\mathbb{N}:n$ is neither a multiple of 3 nor a multiple of 4$\}$. Then the number of elements in $A$ is

26. If the sides of a triangle are in G.P., and its largest angle is twice the smallest, then the common ratio $r$ satisfies the inequality

If, for a positive integer $n$, the quadratic equation $x(x+1) + (x+1)(x+2) + \ldots + (x+\overline{n-1})(x+n) = 10n$ has two consecutive integral solutions, then $n$ is equal to

In an increasing geometric progression of positive terms, the sum of the second and sixth terms is $\dfrac{70}{3}$ and the product of the third and fifth terms is 49. Then the sum of the 4th, 6th and 8th terms is equal to:

$\lim_{n\to\infty}\dfrac{\sum_{k=1}^{n-1}(k-1)(nk-k^2)}{2\sum_{r=1}^n r^3-\sum_{s=0}^n(s^2+(n-s)^2)}=t$. Then $[43t]=$

If $a_1,a_2,\ldots$ are in GP with common ratio $r$, and $\sum_{k=1}^{\infty}a_k=S$, then $\sum_{k=1}^{\infty}a_k^3=$

Using AM $\geq$ GM, if $p, q > 0$, then the maximum value of $p + q$ given $\dfrac{p^2 + q^2}{2} \geq \sqrt{p^2 q^2}$ is:

Sum of infinite number of terms of G.P. is 20 and sum of their squares is 100. The common ratio of G.P. is

Let $\{a_n\}$ and $\{b_n\}$ be two different sequences such that $a_{n+1}=2+a_n$ $\forall n\in\mathbb{N}$, $a_{10}=21$ and $b_n=\dfrac{1}{a_n a_{n+2}}$ $\forall n\in\mathbb{N}$. Then the value of $15\sum_{r=1}^{\infty}b_r$ is

Statement-1: The sum of the series $1 + (1 + 2 + 4) + (4 + 6 + 9) + (9 + 12 + 16) + \ldots + (361 + 380 + 400)$ is 8000.Statement-2: For any natural number $n$, $\sum_{k=1}^{n} k^3 = \left[\frac{n(n+1)}{2}\right]^2$

If a, h, a are in G.P., then a4 - h7 is:

The sum of the series $\dfrac{x}{1-x^2} + \dfrac{x^2}{1-x^4} + \dfrac{x^4}{1-x^8} + \cdots$ to infinite terms, if \(|x|

AP,, is: 10.

Concentric circles of radii 1, 2, 3, …, 100 cm are drawn. The interior of the smallest circle is colored red and the angular regions are colored alternately green and red, so that no two adjacent regions are of the same color. Then, the total area of the green regions in sq. cm is equal to

Given series is $\left(\dfrac{3}{4}\right)^3 + \left(1\dfrac{1}{2}\right)^3 + \left(2\dfrac{1}{4}\right)^3 + 3^3 + \left(3\dfrac{3}{4}\right)^3 + \cdots$ and sum of the first 15 terms of the series is equal to $225k$. Find the value of $k$.

Find three-digit numbers that are divisible by 5 as well as 9 and whose consecutive digits are in AP.

Let an, n € N is an A.P. with common difference 'd' and all whose terms are non-zero. If n 12. If x € R, the numbers (5 +5"), am (2 5 +25") form an A.P. then ‘a’ must lie in the interval approaches infinity, then the sum + will approach

Let a be the first term and r be the common ratio of a GP. If ar2 and ar4 are two terms of the GP whose product is 25, and ar + ar3 = $\frac{25}{2}$, find the sum of three consecutive terms of the form ar3, ar5, ar7.

139. If $T_n$ denotes the $n^{\text{th}}$ term of an arithmetic progression such that $T_p=\dfrac{1}{q}$ and $T_q=\dfrac{1}{p}$, then which of the given options is necessarily a root to the equation $(p+2q-3r)x^2+(q+2r-3p)x+(r+2p-3q)=0$, given that $p+2q-3r\neq 0$?

The value of $e^{(x-1) - \frac{1}{2}(x-1)^2 + \frac{(x-1)^3}{3} - \frac{(x-1)^4}{4} + \cdots}$ is:

For Problems 16–18: There are two sets $A$ and $B$ each of which consists of three numbers in A.P. whose sum is 15 and where $D$ and $d$ are the common differences such that $D - d = 1$. If $\frac{p}{q} = \frac{7}{8}$, where $p$ and $q$ are the product of the numbers, respectively, and $d > 0$ in the two sets.The sum of the product of the numbers in set $A$ taken two at a time is

The sum of the series $1 + \dfrac{1}{4 \cdot 2!} + \dfrac{1}{16 \cdot 4!} + \dfrac{1}{64 \cdot 6!} + \cdots \infty$ is

Let $ \alpha_n, \beta_n $ be the distinct roots of the equation $ x^2 + (n+1)x + n^2 = 0 $. If \[ \sum_{n=2}^{2021} \frac{1}{(\alpha_n + 1)(\beta_n + 1)} \] can be expressed in the form $ \dfrac{a}{b} $, where $ a $ and $ b $ are positive integers, the value of $ (b - a) $ is:

889. Let $\alpha, \beta$ are the roots of the equation $ax^2 + bx + c = 0$ where $\beta = 4\alpha$ $(\alpha > 0)$. If $3a = 2(c - b)$ and $S = \displaystyle\sum_{r=0}^{\infty} \beta(\alpha^r)$, then find the value of $3S$.

Let $\{A_n\}$ and $\{B_n\}$ be two arithmetic progressions with $\dfrac{A_1}{B_1} = \dfrac{a_1}{b_1} = \dfrac{1-1}{2} = 0$, $\dfrac{A_2}{B_2} = \dfrac{1}{4}$, and $\dfrac{A_3}{B_3} = \dfrac{1}{3}$. Find the value of $\dfrac{a_3+a_5+a_7}{3(b_3+b_9)} + \dfrac{a_4+a_{10}}{2(b_2+b_{10})}$.

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 $a_1, a_2, a_3, \ldots, a_{49}$ be in AP such that $\displaystyle\sum_{k=0}^{12} a_{4k+1} = 416$ and $a_9 + a_{43} = 66$. If $a_1^2 + a_2^2 + \cdots + a_{17}^2 = 140m$, then m is equal to

314. For $d \in \left(0, \frac{\pi}{2}\right)$, if $x = \sum_{n=0}^{\infty} \cos^{2n}\theta$, $y = \sum_{n=0}^{\infty} \sin^{2n}\theta$, $z = \sum_{n=0}^{\infty} \cos^{2n}\theta \cdot \sin^{2n}\theta$, then

25. If $a, b$ and $c$ are in A.P. and $b - a, c - b$ and $a$ are in G.P., then $a:b:c$ is

Let $S$ be infinite sum of the series $2 + 3\cos x + 4\cos^2 x + 5\cos^3 x + \ldots\infty$, where $x$ satisfies the equation $|5\cos x + 4| + |5\cos x - 2| = 6$. If the least value of $S$ is equal to $\left(\dfrac{a}{b}\right)$ where $a$ and $b$ are co-prime numbers, then find the value of $(a + b)$.

If $(10)^9 + 2(11)^1(10)^8 + 3(11)^2(10)^7 + \cdots + 10(11)^9 = k(10)^9$, then $k$ is equal to

For $x\geq0$, the least value of $K$, for which $4^{1+x}+4^{1-x}$, $\dfrac{K}{2}$, $16^x+16^{-x}$ are three consecutive terms of an A.P., is equal to:

21. ABC is a right-angled triangle in which $\angle B = 90°$ and $BC = a$. If $n$ points $L_1, L_2, \ldots, L_n$ on $AB$ is divided in $n+1$ equal parts and $L_1M_1, L_2M_2, \ldots, L_nM_n$ are line segments parallel to $BC$ and $M_1, M_2, \ldots, M_n$ are on $AC$, then the sum of the lengths of $L_1M_1, L_2M_2, \ldots, L_nM_n$ is

The value of $E = \sin\dfrac{\pi}{n} + \sin\dfrac{3\pi}{n} + \sin\dfrac{5\pi}{n} + \cdots$ to $n$ terms is:

If $\dfrac{1}{\sqrt{1}+\sqrt{2}}+\dfrac{1}{\sqrt{2}+\sqrt{3}}+\cdots+\dfrac{1}{\sqrt{99}+\sqrt{100}}=m$ and $\dfrac{1}{1\cdot2}+\dfrac{1}{2\cdot3}+\cdots+\dfrac{1}{99\cdot100}=n$, then the point $(m,n)$ lies on the line

If an = (x)1/(2n) + (y)1/(2n) and bn = (x)1/(2n) - (y)1/(2n) for all n ∈ ℕ, then a1a2a3...an is equal to

The sum to the infinity of the series $1 + \dfrac{2}{3} + \dfrac{6}{3^2} + \dfrac{10}{3^3} + \dfrac{14}{3^4} + \cdots$ is

The value of x that satisfies the relation $x = 1 - x + x^2 - x^3 - x^4 - x^5 + \cdots$ to ∞ is

jy|

Given series is $1 + 2 \times 3 + 3 \times 5 + 4 \times 7 + \ldots$ upto $11^{\text{th}}$ term. Find the sum of the series upto the $11^{\text{th}}$ term.

If the sum of the first ten terms of the series $\left(1\frac{3}{5}\right)^2 + \left(2\frac{2}{5}\right)^2 + \left(3\frac{1}{5}\right)^2 + 4^2 + \left(4\frac{4}{5}\right)^2 + \cdots$, is $\frac{16}{5}m$, then $m$ is equal to

Let k be natural number. Defined $ S_k $ as the sum of the infinite geometric series with first term $ (k^2 - 1) $ and common ratio $ \dfrac{1}{k} $, that is $ S_k = \dfrac{k^2-1}{k^0} + \dfrac{k^2-1}{k^1} + \dfrac{k^2-1}{k^2} + \cdots $. The value of $ \displaystyle\sum_{k=1}^{\infty} \dfrac{S_k}{2^{k-1}} $, is:

If the 2nd, 5th and 9th terms of a non-constant A.P. are in G.P., then the common ratio of this G.P. is

If A.M., G.M., and H.M. of the first and last terms of the series 100, 101, 102, …, $n-1, n$ are the terms of the series itself, then the value of $n$ is (\(100

If three non-zero distinct real numbers form an arithmetic progression and the squares of these numbers taken in the same order constitute a geometric progression. Find the sum of all possible common ratios of the geometric progression.

Find the sum of the series $31^3 + 32^3 + \cdots + 50^3$.

For any three positive real numbers $a$, $b$ and $c$, $9(25a^2 + b^2) + 25(c^2 - 3ac) = 15b(3a + c)$. Then

The 15th term of the series $2\dfrac{1}{2} + 1\dfrac{7}{13} + 1\dfrac{1}{9} + \dfrac{20}{23} + \cdots$ is

Let $f(x)$ be a polynomial function of second degree. If $f(1) = f(-1)$ and $a, b, c$ are in AP, then $f'(a)$, $f'(b)$ and $f'(c)$ are in

If H1, H2, ..., Hn be n harmonic means between a and b, then $\frac{\frac{1}{H_1} - a}{\frac{1}{H_n} - b}$ is

Sequences & Series Questions (847)