A farmer built a fence around his square plot. He used 27 fence poles on each side of the square. How many poles did he need altogether ?

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

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.

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

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