Sequences & Series Questions - JEE

Find the value of \[\frac{\displaystyle\sum_{r=1}^{n} \frac{1}{r}}{\displaystyle\sum_{k=1}^{n} \frac{k}{(2n-2k+1)(2n-k+1)}}\]

$\lim_{n \to \infty} \sum_{r=1}^{n} \dfrac{r}{1 \times 3 \times 5 \times 7 \times 9 \times \cdots \times (2r+1)}$ is equal to

Three non-zero numbers a, b, and c are in A.P. Increasing a by 1 or increasing c by 2, the numbers are in G.P. Then find b.

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 first 2000 terms is

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 sum of all numbers is

The next term of the G.P. $x, x^2 + 2,$ and $x^3 + 10$ is

The sum of first nine terms of the series $\dfrac{1^3}{1} + \dfrac{1^3 + 2^3}{1 + 3} + \dfrac{1^3 + 2^3 + 3^3}{1 + 3 + 5} + \cdots$ is

If $a_1, a_2, a_3 \ldots a_n$ are in H.P. and $f(k) = \left(\sum_{r=1}^{n} a_r\right) - a_k$, then $\dfrac{a_1}{f(1)}, \dfrac{a_2}{f(2)}, \dfrac{a_3}{f(3)}, \ldots, \dfrac{a_n}{f(n)}$ are in

If $a, b, c$ are in A.P., then $\dfrac{a}{bc}, \dfrac{1}{c}, \dfrac{2}{b}$ will be in

If $a, b$ and $c$ are in G.P., then $a + b, 2b$ and $b + c$ are in

If $x, a$ and $b$ are in A.P., $a, y$ and $b$ are in G.P., and $a, z, b$ are in H.P. such that $x = 9z$ and $a > 0, b > 0$, then

The sum of series $\displaystyle\sum_{r=0}^{n} (-1)^r (n + 2r)^2$ (where $n$ is even) is

Let $S_n$ denote the sum of first $n$ terms of an A.P. If $S_{2n} = 3S_n$, then find the ratio $S_{3n}/S_n$.

If $a_1, a_2, a_3, \ldots$ are in A.P., then $a_p, a_q, a_r$ are in A.P. if $p$, $q$, $r$ are in

Let $\alpha, \beta \in \mathbb{R}$. If $\alpha, \beta^2$ are the roots of quadratic equation $x^2 - px + 1 = 0$ and $\alpha^2, \beta$ is the roots of quadratic equation $x^2 - qx + 8 = 0$, then the value of $r$ if $\dfrac{r}{8}$ is the arithmetic mean of $p$ and $q$, is

If the sum to infinity of the series $3 + (3 + d)\dfrac{1}{4} + (3 + 2d)\dfrac{1}{4^2} + \cdots \infty$ is $\dfrac{44}{9}$, then find $d$.

How many triplets (a, b, c) exist such that a, b, c, b + c – a, c + a – b, a + b – c, and a + b + c form 7 term arithmetic progression in some order. Here (a, b, c) are distinct positive real numbers.

24. If $x, 2y, 3z$ are in A.P., where the distinct numbers $x, y, z$ are in G.P., then the common ratio of the G.P. is

38. If $x, y$ and $z$ are distinct prime numbers, then

The value of $0.2^{\log_{\sqrt{5}}\!\left(\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\cdots\right)}$ is

The positive integer n for which $2 \times 2^2 + 3 \times 2^3 + 4 \times 2^4 + \cdots + n \times 2^n = 2^{n+10}$ is

Find the sum of infinite series \[\frac{1}{1\times3\times5}+\frac{1}{3\times5\times7}+\frac{1}{5\times7\times9}+\cdots\]

The value of $\displaystyle\sum_{r=1}^{\infty}\frac{8r}{4r^4+1}$ is

If $\displaystyle\sum_{r=1}^n T_r = \dfrac{n(n+1)(n+2)(n+3)}{12}$, where $T_r$ denotes the $r$-th term, then the value of $\displaystyle\lim_{n\to\infty}\sum_{r=1}^n \dfrac{1}{T_r}$ is

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

If the expansion in powers of x of the function $\dfrac{1}{(1-ax)(1-bx)}$ is $a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \cdots$, then $a^n$ is

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

If $\dfrac{a^n + b^n}{a^{n-1} + b^{n-1}}$ is the A.M. between a and b, then find the value of n.

If $a, b$ and $c$ are in A.P., $p, q$ and $r$ are in H.P., and $ap, bq, cr$ are in G.P., then $\dfrac{p}{r} + \dfrac{r}{p}$ is equal to

Let \[S = \frac{\sqrt{1}}{1+\sqrt{1}+\sqrt{2}}+\frac{\sqrt{2}}{1+\sqrt{2}+\sqrt{3}}+\frac{\sqrt{3}}{1+\sqrt{3}+\sqrt{4}}+\cdots+\frac{\sqrt{n}}{1+\sqrt{n}+\sqrt{n+1}}=10\] Then find the value of $n$.

If $a$, $b$, $c$ are positive integers forming an increasing GP, $b-a$ is a perfect cube, and $\log_6 a + \log_6 b + \log_6 c = 6$, then $a+b+c$ is equal to

$1+\dfrac{2}{3}+\dfrac{2\cdot5}{3\cdot6}+\dfrac{2\cdot5\cdot8}{3\cdot6\cdot9}+\cdots$

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

Let $x,y,z$ be three natural numbers such that $x+y+z=10$. The maximum possible value of $xyz+xy+yz+zx$ is

Let one AM $a$ and two GMs $g_1$ and $g_2$ be inserted between $b$ and $c$. Then $\dfrac{g_1^3 + g_2^3}{abc} =$

Let x, y, z be positive real numbers such that $x+y+z=12$ and $x^3y^4z^5=(0.1)(600)^3$. Then $x^3+y^3+z^3$ is equal to

Given $a$, $b$ and $c$ be three distinct real numbers in G.P. and $a + b + c = xb$. Let $r$ be the common ratio of the G.P. If $x$ can take values, then which of the following is correct?

Let $S$ denote the sum of an infinite geometric sequence with $S > 0$. If the second term of this sequence is 1, then the minimum possible value of $S$ is:

Let $\alpha,\beta$ be roots of $x^2-10x+2=0$. Value of $\dfrac{\alpha^{2028}+\beta^{2028}+8\alpha^{2022}+8\beta^{2022}}{\alpha^{2025}+\beta^{2025}}$ is

If a, b, and c also represent the sides of a triangle, then the complete set of a2 is

Let a ∈ (0, 1] satisfies the equation $a^{2008} - 2a + 1 = 0$ and $S = 1 + a + a^2 + \ldots + a^{2007}$. The sum of all possible value(s) of $S$ is

The number of three-term increasing geometrical progressions comprising distinct natural numbers less than or equal to 100, with common ratio as a natural number, is

$\frac{1}{\sqrt{2}+\sqrt{5}}+\frac{1}{\sqrt{5}+\sqrt{8}}+\frac{1}{\sqrt{8}+\sqrt{11}}+\cdots$ $n$ terms is equal to

$\displaystyle\sum_{n=1}^{\infty}\dfrac{4n}{(4n^2-1)^2}=$

Let $a_1,a_2,a_3,\ldots$ be a G.P. of increasing positive terms such that $a_2\cdot a_3\cdot a_4=64$ and $a_1+a_3+a_5=\dfrac{813}{7}$. Then $a_3+a_5+a_7$ is equal to:

If the sum of first 10 terms of an A.P. is 4 times the sum of its first 5 terms, then find the ratio of first term and common difference.

Find the sum $1 + 2\left(1 + \dfrac{1}{50}\right) + 3\left(1 + \dfrac{1}{50}\right)^2 + \cdots$ 50 terms.

If three positive real numbers a, b, c are in A.P. such that $abc = 4$, then the minimum value of b is

Given a, b and c be the 7th, 11th and 13th terms respectively of a non-constant A.P. Let A be the first term and D be the common difference of A.P. If a, b, c are in G.P., find $\dfrac{a}{c}$.

The sum of the first four terms of an A.P. is 56. The sum of the last four terms is 112. If its first term is 11, then find the number of terms.

Sequences & Series Questions (847)