Special series sums

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?

$\displaystyle\lim_{n\to\infty}\tan\left\{\sum_{r=1}^n\tan^{-1}\left(\frac1{1+r+r^2}\right)\right\}$ is equal to _______.

Let the function $f:[0,1]\to\mathbb R$ be defined by

$$f(x) = \frac{4^x}{4^x + 2}.$$

Then the value of

$$f\left(\frac1{40}\right) + f\left(\frac2{40}\right) + f\left(\frac3{40}\right) + \cdots + f\left(\frac{39}{40}\right) - f\left(\frac12\right)$$

is ______

Let $\alpha$ and $\beta$ be the roots of $x^2 - x - 1 = 0$, with $\alpha > \beta$. For all positive integers $n$, define

$$a_n = \frac{\alpha^n - \beta^n}{\alpha - \beta},\quad n \ge 1,$$

$$b_1 = 1\quad\text{and}\quad b_n = a_{n-1} + a_{n+1},\quad n \ge 2.$$

Then which of the following options is/are correct?

For non-negative integers $n$, let

$$f(n) = \frac{\displaystyle\sum_{k=0}^n\sin\left(\frac{k+1}{n+2}\pi\right)\sin\left(\frac{k+2}{n+2}\pi\right)}{\displaystyle\sum_{k=0}^n\sin^2\left(\frac{k+1}{n+2}\pi\right)}$$

Assuming $\cos^{-1}x$ takes values in $[0,\pi]$, which of the following options is/are correct?

The value of

$$\sec^{-1}\left(\frac14\sum_{k=0}^{10}\sec\left(\frac{7\pi}{12} + \frac{k\pi}2\right)\sec\left(\frac{7\pi}{12} + \frac{(k+1)\pi}2\right)\right)$$

in the interval $\left[-\frac\pi4, \frac{3\pi}4\right]$ equals ______

$a_{12} =$

If, for a positive integer n, the quadratic equation,
$$x(x+1)+(x+1)(x+2)+\dots+(x+\overline{n-1})(x+n)=10n$$
has two consecutive integral solutions, then n is equal to :

Let a, b, c ∈ ℝ. If $f(x)=ax^2+bx+c$ is such that $a+b+c=3$ and $f(x+y)=f(x)+f(y)+xy$, ∀ x, y ∈ ℝ, then $\displaystyle\sum_{n=1}^{10}f(n)$ is equal to :

Let $S_n=\dfrac1{1^3}+\dfrac{1+2}{1^3+2^3}+\dfrac{1+2+3}{1^3+2^3+3^3}+\dots+\dfrac{1+2+\dots+n}{1^3+2^3+\dots+n^3}$. If $100S_n=n$, then n is equal to :

If $\displaystyle\lim_{n\to\infty}\frac{1^a+2^a+\dots+n^a}{(n+1)^{a-1}[(na+1)+(na+2)+\dots+(na+n)]}=\frac1{60}$ for some positive real number a, then a is equal to :

The value of $\displaystyle\sum_{k=1}^{13}\frac{1}{\sin\left(\frac{\pi}{4}+\frac{(k-1)\pi}{6}\right)\sin\left(\frac{\pi}{4}+\frac{k\pi}{6}\right)}$ is equal to

If the sum of the first ten terms of the series $\left(1\dfrac35\right)^2+\left(2\dfrac25\right)^2+\left(3\dfrac15\right)^2+4^2+\left(4\dfrac45\right)^2+\dots$, is $\dfrac{16}5m$, then m is equal to :

The value of $\displaystyle\sum_{r=1}^{15}r^2\left(\frac{{}^{15}C_r}{{}^{15}C_{r-1}}\right)$ is equal to :

The sum $\displaystyle\sum_{r=1}^{10}(r^2+1)\times(r!)$ is equal to :

Let α and β be the roots of equation $x^2-6x-2=0$. If $a_n=\alpha^n-\beta^n$, for $n\ge1$, then the value of $\dfrac{a_{10}-2a_8}{2a_9}$ is equal to :

The sum of first 9 terms of the series
$$\frac{1^3}1+\frac{1^3+2^3}{1+3}+\frac{1^3+2^3+3^3}{1+3+5}+\dots$$
is :

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

If $\displaystyle\sum_{n=1}^5\frac1{n(n+1)(n+2)(n+3)}=\frac k3$, then k is equal to :

If the sum
$$\frac{3}{1^2}+\frac{5}{1^2+2^2}+\frac{7}{1^2+2^2+3^2}+\dots\text{ upto 20 terms}$$
is equal to $\dfrac{k}{21}$, then $k$ is equal to :