Sequences and Series

Let the sum of an infinite G.P., whose first term is $a$ and the common ratio is $r$, be 5. Let the sum of its first five terms be $\dfrac{98}{25}$. Then the sum of the first 21 terms of an AP, whose first term is $10ar$, $n^{\text{th}}$ term is $a_n$ and the common difference is $10ar^2$, is equal to :

$\dfrac{2^3-1^3}{1\times7}+\dfrac{4^3-3^3+2^3-1^3}{2\times11}+\dfrac{6^3-5^3+4^3-3^3+2^3-1^3}{3\times15}+\cdots+\dfrac{30^3-29^3+28^3-27^3+\ldots+2^3-1^3}{15\times63}$ is equal to ______.

Let $\alpha$, $\beta$ and $\gamma$ be three positive real numbers. Let $f(x)=\alpha x^5+\beta x^3+\gamma x$, $x\in\mathbf R$ and $g:\mathbf R\to\mathbf R$ be such that $g(f(x))=x$ for all $x\in\mathbf R$. If $a_1,a_2,a_3,\ldots,a_n$ be in arithmetic progression with mean zero, then the value of $f\left(g\left(\dfrac1n\displaystyle\sum_{i=1}^nf(a_i)\right)\right)$ is equal to :

Consider the sequence $a_1,a_2,a_3,\ldots$ such that $a_1=1$, $a_2=2$ and $a_{n+2}=\dfrac2{a_{n+1}}+a_n$ for $n=1,2,3,\ldots$. If

$$\left(\frac{a_1+\frac1{a_2}}{a_3}\right)\cdot\left(\frac{a_2+\frac1{a_3}}{a_4}\right)\cdot\left(\frac{a_3+\frac1{a_4}}{a_5}\right)\cdots\left(\frac{a_{30}+\frac1{a_{31}}}{a_{32}}\right)=2^\alpha\left({}^{61}C_{31}\right),$$

then $\alpha$ is equal to :

For $p,q\in\mathbf R$, consider the real valued function $f(x)=(x-p)^2-q$, $x\in\mathbf R$ and $q>0$. Let $a_1$, $a_2$, $a_3$ and $a_4$ be in an arithmetic progression with mean p and positive common difference. If $|f(a_i)|=500$ for all $i=1,2,3,4$, then the absolute difference between the roots of $f(x)=0$ is ______.

Let $x_1,x_2,x_3,\ldots,x_{20}$ be in geometric progression with $x_1=3$ and the common ratio $\dfrac12$. A new data is constructed replacing each $x_i$ by $(x_i-i)^2$. If $\bar x$ is the mean of new data, then the greatest integer less than or equal to $\bar x$ is ______.

Let the coefficients of the middle terms in the expansion of $\left(\dfrac1{\sqrt6}+\beta x\right)^4$, $(1-3\beta x)^2$ and $\left(1-\dfrac\beta2x\right)^6$, $\beta>0$, respectively form the first three terms of an A.P. If d is the common difference of this A.P., then $50-\dfrac{2d}{\beta^2}$ is equal to ______.

If $\dfrac6{3^{12}}+\dfrac{10}{3^{11}}+\dfrac{20}{3^{10}}+\dfrac{40}{3^9}+\ldots+\dfrac{10240}3=2^n\cdot m$, where m is odd, then m · n is equal to ______.

If $\dfrac1{(20-a)(40-a)}+\dfrac1{(40-a)(60-a)}+\dots+\dfrac1{(180-a)(200-a)}=\dfrac1{256}$, then the maximum value of a is :

Let $a_1,a_2,a_3,\dots$ be an A.P. If $\displaystyle\sum_{r=1}^\infty\frac{a_r}{2^r}=4$, then $4a_2$ is equal to _______.

If $\dfrac1{2\times3\times4}+\dfrac1{3\times4\times5}+\dfrac1{4\times5\times6}+\dots+\dfrac1{100\times101\times102}=\dfrac k{101}$, then 34 k is equal to _______.

Let $\{a_n\}_{n=0}^\infty$ be a sequence such that $a_0=a_1=0$ and $a_{n+2}=3a_{n+1}-2a_n+1$, ∀ n ≥ 0.

Then $a_{25}a_{23}-2a_{25}a_{22}-2a_{23}a_{24}+4a_{22}a_{24}$ is equal to

$\displaystyle\sum_{r=1}^{20}(r^2+1)(r!)$ is equal to

The number of elements in the set $S=\left\{x\in\mathbb R:2\cos\left(\dfrac{x^2+x}6\right)=4^x+4^{-x}\right\}$ is :

Let α, β (α > β) be the roots of the quadratic equation $x^2-x-4=0$. If $P_n=\alpha^n-\beta^n$, n ∈ ℕ, then $\dfrac{P_{15}P_{16}-P_{14}P_{16}-P_{15}^2+P_{14}P_{15}}{P_{13}P_{14}}$ is equal to _______.

The coefficient of $x^{1011}$ in the expression $(1+2x)^{2022}+2x(1+2x)^{2021}+(2x)^2(1+2x)^{2020}+\dots+(2x)^{2022}$ is

For |α| ≥ 1, let $5^{4-2\alpha}$, 63, $5^{2\alpha-1}$ be in A.P. with the common difference d. Let S$_n$ denote the sum of the first n terms of an A.P., whose first term is α$^2$ and the common difference is d. If S$_{30}$ − S$_{15}$ = 30k, then k is equal to

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

Let $A_k=\displaystyle\int_{k\pi}^{(k+1)\pi}e^{-x}\sin x\,dx$, k = 0, 1, 2, ..., 20. Then $\displaystyle\sum_{k=0}^{20}|A_k|$ is equal to

For any positive integer $n$, let $S_n: (0, \infty) \to \mathbb R$ be defined by
$$S_n(x) = \sum_{k=1}^n \cot^{-1}\left(\frac{1 + k(k+1)x^2}{x}\right),$$
where for any $x \in \mathbb R$, $\cot^{-1}(x) \in (0, \pi)$ and $\tan^{-1}(x) \in \left(-\frac\pi2, \frac\pi2\right)$. Then which of the following statements is (are) TRUE?