Sequences & Series Questions - JEE

If $P(n) = \dfrac{1}{n+1} + \dfrac{1}{n+2} + \cdots + \dfrac{1}{2n} > \dfrac{13}{24}$, then $P(n)$ is true for $n \geq$ __________.

a > (2) 27 (3) 16 (4)7 = co 1)" 23. If A=) = , then is equal to: [EE (Main) 2022]

The value of $\displaystyle\sum_{r=1}^{15} r^2 \left(\frac{{}^{15}C_r}{{}^{15}C_{r-1}}\right)$ is equal to

Find the sum $1^2 + (1^2 + 2^2) + (1^2 + 2^2 + 3^2) + \cdots$ up to the 22nd term.

If the sum of $n$ terms of an A.P. is $cn(n-1)$, where $c \neq 0$, then the sum of the squares of these terms is

Statement-1: The sum of the series $1 + (1+2+4) + (4+6+9) + (9+12+16) + \cdots + (361+380+400) = 8000$.Statement-2: $\displaystyle\sum_{k=1}^{n}(k^3 - (k-1)^3) = n^3$ for any natural number $n$.

If $H_1, H_2, \ldots, H_{20}$ are 20 harmonic means between 2 and 3, then $\dfrac{H_1 + 2}{H_1 - 2} + \dfrac{H_{20} + 3}{H_{20} - 3} =$

Fifth term of a G.P. is 2, then the product of its 9 terms is

Given that $ar^{n+1} = ar^n + ar^{n-1}$, find the value of the common ratio $r$.

Suppose q is the first of n harmonic means between two positive numbers a and b. The value of q is

When the ninth term of an AP is divided by its second term, we get 5 as the quotient. When the thirteenth term is divided by the sixth term, the quotient is 2 and the remainder is 5. Find the second term of the AP.

If A1, A2, A3, ..., Am are arithmetic means between −3 and 828, and the sum of these arithmetic means equals 14025, find the value of m.

Find the first negative term of the sequence $20, 19\frac{3}{4}, 19\frac{1}{2}, ...$

There are two numbers a and b whose product is 192 and the quotient of AM by HM of their greatest common divisor and least common multiple is $\frac{169}{48}$. The smaller of a and b is

If a, b, c are non-zero real numbers, then the minimum value of the expression\[\frac{(a^8 + 4a^4 + 1)(b^4 + 3b^2 + 1)(c^2 + 2c + 2)}{a^4b^2}\]equals

Example 25 (Statement-1): Let \[\sum_{r=2}^{n-1} \frac{(r+1)\binom{n}{2}}{\binom{2}{2}\binom{(n!)}{4n-2}} = \sum_{r=2}^{n-1} \frac{(r+1)\binom{n!}{3n-2n}}{\binom{(n!)}{3n-2n}}\] then $5\sum r = 0$.Statement-2: $5\sum_{r=2}^{n-1} r = \sum_{r=2}^{n-1} 2 + \sum_{r=2}^{n-1} 3 + \sum_{r=2}^{n-1} 4 + \cdots + \sum_{r=2}^{n-1}$

$\sum_{r=0}^{30} P(r)$ is equal to:

Given an AP whose terms are all positive integers. The sum of its first nine terms is greater than 200 and less than 220. If the second term in it is 12, then its 4th term is

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

If the value of $\sum_{i=0}^{n} \sum_{j=0}^{n} \sum_{k=0}^{n} \frac{1}{3^{i+j+k}}$ ($i \neq j \neq k$) is equal to $\frac{m}{n}$, where $m, n$ are coprime natural numbers, then $m + n$ is equal to ____.

Integers $1, 2, 3, ..., n$ where $n > 2$ are written on a board. Two numbers $m, k$ such that $1 < m < n, 1 < k < n$ are removed and the average of the remaining numbers is found to be 17. What is the maximum sum of the two removed numbers ?

The sum of n terms of the series \[1.4 + 3.04 + 5.004 + 7.0004 + \ldots\] is

Let $x_1, x_2, x_3, ..., x_{2018}$ be real numbers different from 1, such that $x_1 + x_2 + ... + x_{2018} = 1$ and $\frac{x_1}{1-x_1} + \frac{x_2}{1-x_2} + ... + \frac{x_{2018}}{1-x_{2018}} = 1$ then the value of $\frac{x_1^2}{1-x_1} + \frac{x_2^2}{1-x_2} + ... + \frac{x_{2018}^2}{1-x_{2018}}$ is equal to ____.

Let $x_1, x_2, ..., x_{2018}$ be positive real numbers such that $x_1 + x_2 + ... + x_{2018} = 1$. Determine the smallest constant $k$ such that $k \sum_{i=1}^{2018} \frac{x_i^2}{1-x_i} \geq 1$

Let $x, y, z$ are positive real numbers satisfy $2x - 2y + \frac{1}{z} = \frac{1}{2018}, 2y - 2z + \frac{1}{x} = \frac{1}{2018}, 2z - 2x + \frac{1}{y} = \frac{1}{2018}$ then $x + y - z$ is equal to ____.

Let $a, b, c$ be positive real number such that $a + b + c \geq 4$, then find the minimum value of $\frac{a^3}{(a-b)(a-c)} + \frac{b^3}{(b-c)(b-a)} + \frac{c^3}{(c-a)(c-b)}$.

A total prize of ₹8000 is to be distributed among 16 teams. If the first place team gets ₹275 and there is a common difference of prize amounts between consecutive places, find the prize for the team that comes first.

Let $S_n, S_{2n}, S_{3n}$ are respectively the sums of first $n$, $2n$, $3n$ terms of an arithmetic progression, then $S_{3n} =$

The sum of infinite series $1 - \frac{2^2}{5} - \frac{3^2}{5^2} - \frac{4^2}{5^3} - \frac{5^2}{5^4} - \frac{6^2}{5^5} - \ldots$ is equal to:

The sum of first 20 terms of the sequence 0.7, 0.77, 0.777, …, is

If $(3x - 1), (3x - 3)$ are the first three terms of an arithmetic progression, then the sum of the first five terms can be

Let the $r$-th term, $t_r$ of a series is given by $t_r = \frac{r}{1 + r^2 + r^4}$. The value of $\lim_{n \to \infty} \sum_{r=1}^{n} t_r$ is

The positive integer $n$, for which the solutions of the equation $x(x+2)+(x+2)(x+4)+\cdots+(x+2n-2)(x+2n)=\dfrac{8n}{3}$ are two consecutive even integers, is:

$\dfrac{6}{3^{26}}+\dfrac{10\cdot1}{3^{25}}+\dfrac{10\cdot2}{3^{24}}+\dfrac{10\cdot2^2}{3^{23}}+\cdots+\dfrac{10\cdot2^{24}}{3}$ is equal to:

If $\displaystyle\sum_{r=1}^{25}\left(\dfrac{r}{r^4+r^2+1}\right)=\dfrac{p}{q}$, where $p$ and $q$ are positive integers such that $\gcd(p,q)=1$, then $p+q$ is equal to _____.

Suppose $a,b,c$ are in A.P. and $a^2,2b^2,c^2$ are in G.P. If $a<b<c$ and $a+b+c=1$, then $9(a^2+b^2+c^2)$ is equal to _____.

If the sum of the first four terms of an A.P. is 6 and the sum of its first six terms is 4, then the sum of its first twelve terms is

The value of $\displaystyle\sum_{k=1}^\infty(-1)^{k+1}\left(\dfrac{k(k+1)}{k!}\right)$ is

Let $a_1=1$ and for $n\geq1$, $a_{n+1}=\dfrac{1}{2}a_n+\dfrac{n^2-2n-1}{n^2(n+1)^2}$. Then $\left|\displaystyle\sum_{n=1}^\infty\left(a_n-\dfrac{2}{n^2}\right)\right|$ is equal to _____.

In a G.P., if the product of the first three terms is 27 and the set of all possible values for the sum of its first three terms is $\mathbb{R}-(a,b)$, then $a^2+b^2$ is equal to _____.

Let $729,81,9,1,\ldots$ be a sequence and $P_n$ denote the product of the first $n$ terms of this sequence. If $2\displaystyle\sum_{n=1}^{40}(P_n)^{1/n}=\dfrac{3^\alpha-1}{3^\beta}$ and $\gcd(\alpha,\beta)=1$, then $\alpha+\beta$ is equal to

If the (m + 1)th, (n + 1)th and (r + 1)th terms of an A.P. are in G.P. and m, n, r are in H.P., then the ratio of the common difference to the first term in the A.P. is equal to

Consider two sets A and B, each containing three numbers in A.P. Let the sum and the product of the elements of A be 36 and p respectively and the sum and the product of the elements of B be 36 and q respectively. Let d and D$p+q$be the common differences of AP's in A and B respectively such that$D = d + 3$,$d > 0.$If$p-q = 19$5, then$p - q$is equal to

If $T_r = \sqrt{r}\sqrt{r+1}\left(\dfrac{4r+5}{(r+2)+\sqrt{r^2-1}}\right)$, then find the value of $\dfrac{1}{\sqrt{68}}\sum_{r=1}^{16} T_r$.

Ex. 83: Sum of all values of x satisfying the equation x = $ \sqrt[]{4 + \sqrt[]{4 + \sqrt[]{4 + \cdots}}} $ is

If A1, A2 be A.M.'s, G1, G2 be G.M.'s and H1, H2 be H.M.'s between two numbers, then

Let $2^{\text{nd}},8^{\text{th}}$ and $44^{\text{th}}$ terms of a non-constant A.P. be respectively the $1^{\text{st}},2^{\text{nd}}$ and $3^{\text{rd}}$ terms of a G.P. If the first term of the A.P. is 1, then the sum of first 20 terms is equal to

The $20^{\text{th}}$ term from the end of the progression $20,19\dfrac{1}{4},18\dfrac{1}{2},17\dfrac{3}{4},\ldots,-129\dfrac{1}{4}$ is:

If in a G.P. of 64 terms, the sum of all the terms is 7 times the sum of the odd terms of the G.P., then the common ratio of the G.P. is equal to

The sum$1 + 3 + 11 + 25 + 45 + 71+..$upto 20 terms, is equal to

Sequences & Series Questions (847)