Answer
a) $2550$
b) $n(n+1)$
Work Step by Step
a) By applying the same logic Gauss applied, we notice that for the sum
$2+4+\ldots+100$ we can group the numbers from $2$ to $100$ into $25$ pairs
of the form
$2+100=102$
$4+98=102$
$\cdot$
$\cdot$
$\cdot$
$50+52=102$
The total sum is $25 \cdot 102=2550$
b) We apply the same logic for the sum $2+4+...+2n$.
$2+2n=2+2n$
$4+(2n-2)=2+2n$
......
$2n+2=2n+2$
The total sum $S$ is:
$2S=n\cdot (2n+2)$
$S=\frac{2n(n+1)}{2}=n(n+1)$
