Sequences & Series Questions - JEE

If $\{a_i\}_{i=1}^n$ where $n$ is an even integer is an AP with common difference 1, and $\sum_{i=1}^n a_i=192$, $\sum_{i=1}^{n/2} a_{2i}=120$, then $n$ is equal to

If $f(x) + f(\alpha x) + f(\alpha^2 x) + \cdots + f(\alpha^6 x) = 7A_0 + \sum_{k=1}^{20} A_k x^k \left(1 + \alpha^k + \cdots + \alpha^{6k}\right)$, where $\alpha$ is a 7th root of unity and $\alpha \neq 1$, then for $k \neq 7$ and $k \neq 14$, $1 + \alpha^k + \alpha^{2k} + \cdots + \alpha^{6k} = 0$. Hence $f(x) + f(\alpha x) + \cdots + f(\alpha^6 x) = 7(A_0 + A_7 x^7 + A_{14} x^{14})$. What is the value of $k$ such that the expression equals $7(A_0 + A_7 x^7 + A_{14} x^{14})$?

$\dfrac{2^3-1^3}{1\times7}+\dfrac{4^3-3^3+2^3-1^3}{2\times11}+\dfrac{6^3-5^3+\cdots+1^3}{3\times15}+\cdots+\dfrac{30^3-29^3+\cdots+1^3}{15\times63}$ is equal to

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.

Sequences & Series Questions (847)