Special series sums

The value of $\cot\left(\displaystyle\sum_{n=1}^{50}\tan^{-1}\left(\frac1{1+n+n^2}\right)\right)$ is

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 $\{a_n\}_{n=0}^\infty$ 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:

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

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 ______

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 ______.

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

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 :

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 :

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

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 _______.

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