If $x, y, z$ are in $\mathrm{A}.\mathrm{P}.$ and $x, y,(z+1)$ are in $G.P.$ then _____.

The angles of a triangle are in AP (arithmetic progression). If measure of the smallest angle is $50°$ less than that of the largest angle, then find the largest angle.

How many numbers are there from$200$ to $800$ which are neither divisible by $5$ nor by $7$?

If $m$ is the $A.M.$of two distinct real numbers $I$ and $n$ $\left(I, n>1\right)$  and $G_1, G_2 \text{and }{G}_3$ are three geometric means between $I \text{and} n$, then $G_1^4+2G_2^4+G_3^4$ equals

$\text{ Let }{S}_n=1·(n-1)+2·(n-2)+3·(n-3)+\dots +(n-1)·1, n⩾4.$ 

$\text{The sum }\sum _{\mathrm{n}=4}^{\infty }\left(\frac{2{ \mathrm{S}}_{\mathrm{n}}}{\mathrm{n}!}-\frac{1}{(\mathrm{n}-2)!}\right)\text{ is equal to :}$