Sequences and Series

The value of $\displaystyle\lim_{n\to\infty}6\tan\left\{\sum_{r=1}^n\tan^{-1}\left(\frac1{r^2+3r+3}\right)\right\}$ is equal to :

Let for $n=1,2,\dots,50$, $S_n$ be the sum of the infinite geometric progression whose first term is $n^2$ and whose common ratio is $\dfrac1{(n+1)^2}$. Then the value of $\dfrac1{26}+\displaystyle\sum_{n=1}^{50}\left(S_n+\frac2{n+1}-n-1\right)$ is equal to ______.

Let $A=[a_{ij}]$ be a square matrix of order 3 such that $a_{ij}=2^{j-i}$, for all $i,j=1,2,3$. Then, the matrix $A^2+A^3+\dots+A^{10}$ is equal to :

Let $\{a_n\}_{n=0}^\infty$ be a sequence such that $a_0=a_1=0$ and $a_{n+2}=2a_{n+1}-a_n+1$ for all $n\ge0$. Then, $\displaystyle\sum_{n=2}^\infty\frac{a_n}{7^n}$ is equal to :

Let $c,k\in\mathbf R$. If $f(x)=(c+1)x^2+\left(1-c^2\right)x+2k$ and $f(x+y)=f(x)+f(y)-xy$, for all $x,y\in\mathbf R$, then the value of $\left|2\left(f(1)+f(2)+f(3)+\dots+f(20)\right)\right|$ is equal to ______.

The sum of the infinite series $1+\dfrac56+\dfrac{12}{6^2}+\dfrac{22}{6^3}+\dfrac{35}{6^4}+\dfrac{51}{6^5}+\dfrac{70}{6^6}+\dots$ is equal to:

Let 3, 6, 9, 12, ... upto 78 terms and 5, 9, 13, 17, ... upto 59 terms be two series. Then, the sum of the terms common to both the series is equal to ______.

Let $M=\begin{bmatrix}0&-\alpha\\\alpha&0\end{bmatrix}$, where $\alpha$ is a non-zero real number an $N=\displaystyle\sum_{k=1}^{49}M^{2k}$. If $\left(I-M^2\right)N=-2I$, then the positive integral value of $\alpha$ is ______.

For two positive real numbers a and b such that $\dfrac1{a^2}+\dfrac1{b^3}=4$, the minimum value of the constant term in the expansion of $\left(ax^{\frac18}+bx^{-\frac1{12}}\right)^{10}$ is :

The value of $1+\dfrac1{1+2}+\dfrac1{1+2+3}+\dots+\dfrac1{1+2+3+\dots+11}$ is equal to :

Let for $f(x)=a_0x^2+a_1x+a_2$, $f'(0)=1$ and $f'(1)=0$. If $a_0,a_1,a_2$ are in an arithmatico-geometric progression, whose corresponding A.P. has common difference 1 and corresponding G.P. has common ratio 2, then $f(4)$ is equal to ______.

Let a, b be two non-zero real numbers. If p and r are the roots of the equation $x^2-8ax+2a=0$ and q and s are the roots of the equation $x^2+12bx+6b=0$, such that $\dfrac1p,\dfrac1q,\dfrac1r,\dfrac1s$ are in A.P., then $a^{-1}-b^{-1}$ is equal to ______.

Let $a_1=b_1=1$, $a_n=a_{n-1}+2$ and $b_n=a_n+b_{n-1}$ for every natural number $n\ge2$. Then $\displaystyle\sum_{n=1}^{15}a_n\cdot b_n$ is equal to ______.

The sum $\displaystyle\sum_{n=1}^{21}\frac3{(4n-1)(4n+3)}$ is equal to

Consider two G.Ps. 2, $2^2$, $2^3$, ... and 4, $4^2$, $4^3$, ... of 60 and n terms respectively. If the geometric mean of all the $60+n$ terms is $(2)^{\frac{225}8}$, then $\displaystyle\sum_{k=1}^nk(n-k)$ is equal to :

The series of positive multiples of 3 is divided into sets : {3}, {6, 9, 12}, {15, 18, 21, 24, 27}, ... Then the sum of the elements in the 11$^{\text{th}}$ set is equal to ______.

If $\displaystyle\sum_{k=1}^{10}\frac{k}{k^4+k^2+1}=\frac mn$, where $m$ and $n$ are co-prime, then $m+n$ is equal to ______

Different A.P.'s are constructed with the first term 100, the last term 199, and integral common differences. The sum of the common differences of all such A.P.'s having at least 3 terms and at most 33 terms is ______.

Suppose $a_1,a_2,\ldots,a_n,\ldots$ be an arithmetic progression of natural numbers. If the ratio of the sum of first five terms to the sum of first nine terms of the progression is 5 : 17 and $110<a_{15}<120$, then the sum of the first ten terms of the progression is equal to

Let $f(x)=2x^2-x-1$ and $S=\{n\in\mathbf Z:|f(n)|\le800\}$. Then, the value of $\displaystyle\sum_{n\in S}f(n)$ is equal to ______.