Question

Prove by method of induction, for all $n\in N$
$2+4+6+....+2n=n\left(n+1\right)$

Answer 1

First, that answer is correct

$2+4+b+...+2n=2(1+2+\mathrm{3....}+n)=\frac{n(n+1)}{2}$

$=n(n+1)$

Second, suppose that is true you have $2+4+b+.....+2n$

$=2\times (n+1)$

That is $S\left(n\right)$ and $2(n+1)$ to both sides giving

$2+4+.....+2n+2(n+1)=n\ast (n+1)+2(n+1)(n+1)$

Which is $S(n+1)$ Hence proved.

that $2+4+b+......+2n=n(n+1)$