Special series sums

If
$$\Delta_r=\begin{vmatrix}r&2r-1&3r-2\\\dfrac n2&n-1&a\\\dfrac12n(n-1)&(n-1)^2&\dfrac12(n-1)(3n+4)\end{vmatrix},$$
then the value of $\displaystyle\sum_{r=1}^{n-1}\Delta_r$ :

Let $f(n)=\left[\dfrac13+\dfrac{3n}{100}\right]n$, where $[n]$ denotes the greatest integer less than or equal to $n$. Then $\displaystyle\sum_{n=1}^{56}f(n)$ is equal to :

The value of $\cot\left(\displaystyle\sum_{n=1}^{23}\cot^{-1}\left(1+\sum_{k=1}^n2k\right)\right)$ is

Let $S_n=\displaystyle\sum_{k=1}^{4n}(-1)^{\frac{k(k+1)}2}k^2$. Then $S_n$ can take value(s)

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

Let $S_k$, $k=1,2,\ldots,100$, denote the sum of the infinite geometric series whose first term is $\dfrac{k-1}{k!}$ and the common ratio is $\dfrac1k$. Then the value of $\dfrac{100^2}{100!}+\displaystyle\sum_{k=1}^{100}\left|(k^2-3k+1)S_k\right|$ is

If the sum of first $n$ terms of an A.P. is $cn^2$, then the sum of squares of these $n$ terms is

Let $g(x)=\log f(x)$ where $f(x)$ is a twice differentiable positive function on $(0,\infty)$ such that $f(x+1)=xf(x)$. Then, for $N=1,2,3,\ldots$,
$$g''\left(N+\frac12\right)-g''\left(\frac12\right)=$$

The sum $V_1+V_2+\cdots+V_n$ is