If $a, b$ and $c$ are distinct positive numbers, not equal to unity. Such that $a b c=1$, then the value of $\log _b a \cdot \log _c a+\log _c b$ $\cdot \log _a b+\log _a c \log _b c$ is

Sum of first ' $n$ ' terms of a series $a_1+a_2+\ldots+a_n$ is given by $S_n=\frac{n\left(n^2-1\right)(n+2)}{4}$, then the value of $\lim _\limits{n \rightarrow \infty} \sum_\limits{r=2}^n \frac{1}{a_r}$ is

The area of circle touching parabola $y=x^2$ at $(1,1)$ and having directrix of $y=x^2$ as its normal is $125 A \pi$, then $A$ is

The sum of the infinite terms of the series $\cot ^{-1}\left(1^2+\frac{3}{4}\right)+\cot ^{-1}\left(2^2+\frac{3}{4}\right)$ $+\cot ^{-1}\left(3^2+\frac{3}{4}\right)+\ldots$ is equal to