Sequences & Series Questions - JEE

The sum $\displaystyle\sum_{n=1}^{\infty}\frac{9n^2}{(3n)!}$ is equal to

The value of $\dfrac{1}{3^2+1}+\dfrac{1}{4^2+2}+\dfrac{1}{5^2+3}+\cdots$ to $\infty$ is

Let $x_1, x_2, \ldots, x_{100}$ be an A.P. with $x_1=1$ and $x_{100}=199$. If $y_i=i(x_i+1)$; $i=1,2,\ldots,100$, then mean of $y_1, y_2, \ldots, y_{100}$ is

The value of $$\sum_{r=1}^{\infty} \frac{4}{4r^4 + 1}$$ is equal to:

The third term of a G.P. is 2. Then the product of the first five terms is

Let $a$ denotes the number of non-negative values of $p$ for which the equation $p2^x + 2^{-x} = 5$ possess a unique solution. If $a, \alpha_1, \alpha_2, \ldots, \alpha_{20}, 6$ are in H.P. and $a, \beta_1, \beta_2, \ldots, \beta_{20}, 6$ are in A.P., find $\alpha_{18}\beta_3$.

(a 2 ws on Ina GP, first term is 1. If 47, + 573 is minimum, then its common ratio is

(a) ) -2 oF (0)

Let {a,, }7_1 be a sequence such that a, = 1, az = 1 and ay42 = 2ay41 + dy for all n >

31. any integer n with 1

The real number x when added to its inverse gives the minimum value of the sum at x equal to

If a, b, c, d are distinct integers in an A.P. such that $d = a^2 + b^2 + c^2$, then find the value of $a + b + c + d$.

Given, common difference > 0 and $2 S_{3n} = S_{4n} - S_{3n}$, where $S_n = Pn^2 + Qn$Find the ratio $\frac{S_{2n}}{S_{4n} - S_{2n}}$

Find the coefficient of $x^{n-2}$ in the expansion of $f(x) = (x-1)(x-2)(x-3)\cdots(x-n)$

The value of $\displaystyle\sum_{n=0}^{\infty} \dfrac{(\ln x)^n}{n!}$ is:

Some identical balls are arranged in rows to form an equilateral triangle. The first row consists of one ball, the second row consists of two balls and so on. If 99 more identical balls are added to the total number of balls used in forming the equilateral triangle, then all these balls can be arranged in a square whose each side contains exactly 2 balls less than the number of balls each side of the triangle contains. Then the number of balls used to form the equilateral triangle is __________.

If $a, G, b$ are in GP and $\dfrac{1}{a}, M, \dfrac{1}{b}$ are in AP, then $M$ equals:

Let $a_1,a_2,a_3,\ldots$ be in an arithmetic progression of positive terms. Let $A_k=a_1^2-a_2^2+a_3^2-a_4^2+\ldots+a_{2k-1}^2-a_{2k}^2$. If $A_3=-153$, $A_5=-435$ and $a_1^2+a_2^2+a_3^2=66$, then $a_{17}-A_7$ is equal to

Let the first term of a series be $T_1=6$ and its $r$th term $T_r=3T_{r-1}+6^r$, $r=2,3,\ldots,n$. If the sum of the first $n$ terms of this series is $\dfrac{1}{5}(n^2-12n+39)(4\cdot6^n-5\cdot3^n+1)$, then $n$ is equal to

If 19th term of a non-zero A.P. is zero, then its (49th term) : (29th term) is

Let the sum of the first three terms of an AP be 39 and the sum of its last four terms be 178. If the first term of this AP is 10, then the median of the AP is

Let α and β be the roots of the equation px2 + qx + r = 0, p ≠ 0. If p, q, r are in AP and $\dfrac{1}{\alpha} + \dfrac{1}{\beta} = 4$, then the value of |α − β| is

If $a$, $b$, $c$ are in AP and $(a+2b-c)(2b+c-a)(c+a-b)=\lambda abc$, then the value of $\lambda$ is

The value of $\displaystyle\sum_{r=16}^{30}(r+2)(r-3)$ is equal to:

If $\left(\dfrac{1}{\alpha+1}+\dfrac{1}{\alpha+2}+\cdots+\dfrac{1}{\alpha+1012}\right)-\left(\dfrac{1}{2\cdot1}+\dfrac{1}{4\cdot3}+\dfrac{1}{6\cdot5}+\cdots+\dfrac{1}{2024\cdot2023}\right)=\dfrac{1}{2024}$, then $\alpha$ is equal to

Let $S_n = c - \dfrac{1}{(n+1)(n+2)(n+3) \cdot 3}$. Find $\lim_{n \to \infty} S_n$ (given that $c = \dfrac{1}{18}$).

If $S_n=0.7+0.77+0.777+\cdots$ to $n$ terms, then $S_n=$

If $S=\displaystyle\sum_{r=1}^\infty\dfrac{r}{4^r}$, then $16S$ is

Let fourth term of an arithmetic progression be 6 and $m^{\text{th}}$ term be 18. If A.P. has integral terms only then the numbers of such A.P.s is ___.

Find the sum $\displaystyle\sum_{j=1}^{n}\sum_{i=1}^{n} i \times 3^j$.

If the sum of $n$ terms of a G.P. is $3 - \dfrac{3^{n+1}}{4^{2n}}$, then find the common ratio.

For Problems 10–12: Four different integers form an increasing A.P. One of these numbers is equal to the sum of the squares of the other three numbers.The common difference of the four numbers is

If $A_1, A_2, G_1, G_2$ and $H_1, H_2$ are two arithmetic, geometric and harmonic means, respectively, between two quantities $a$ and $b$, then $ab$ is equal to

For Problems 22–24: Two consecutive numbers from 1, 2, 3, …, $n$ are removed. The arithmetic mean of the remaining numbers is $\frac{105}{4}$.The value of $n$ lies in

If the sum of $m$ terms of an A.P. is the same as the sum of its $n$ terms, then the sum of its $(m+n)$ terms is

In any A.P., if sum of first six terms is 5 times the sum of next six terms then which term is zero?

$a, b, c, d \in R^+$ such that $a, b$ and $c$ are in A.P. and $b, c$ and $d$ are in H.P., then

If $S_n = nP + \dfrac{n(n-1)}{2}Q$, where $S_n$ denotes the sum of the first $n$ terms of an A.P., then find the common difference.

Find the sum $\displaystyle\sum_{r=1}^{20}\frac{r}{r^4+\dfrac{1}{4}}$.

If first and $(2n-1)^{\text{th}}$ terms of an A.P., G.P., and H.P. are equal and their $n^{\text{th}}$ terms are a, b, c, respectively, then

Let $P(n) = \dfrac{1}{n+1} + \dfrac{1}{n+2} + \cdots + \dfrac{1}{2n} > \dfrac{13}{24}$, ∀n ∈ N. Then P(n) is true for:

$\displaystyle\sum_{r=0}^n(-1)^r\binom{n}{r}\dfrac{1}{r+3}$

16) 2 (2) 2 (3)2 (4)1 P4+243?+.. upto nterms = 25 a then the value of nis [EE (Main) 2023]

Find the greatest value of the product of three positive numbers if the sum of their products taking two at a time is 12.

55 1 1 1 1 (1 22 (2) —— (3) 2& (4) 28. 15 111 111 111 111 The product 2*-44-8*8-16#8- .... to oo is equal to: [JEE (Main) 2020]

If non-zero numbers a, b, c are in HP, then the straight line $\dfrac{x}{a} + \dfrac{y}{b} + \dfrac{1}{c} = 0$ always passes through a fixed point. That point is

The sum of four whole numbers in AP is 24 and their product is 945; find the numbers.

Sequences & Series Questions (847)