Answer
The formula was proved for $n=1$
The formula is correct if $n$ is changed by $n+1$
Work Step by Step
Let's prove the formula for $n=1$:
$1(1+2)=1(3)=3$
It is correct!
Now, suppose that the formula is correct, that is:
$3+5+7+...+2n+1=n(n+2)$
Now, let's prove the formula for $n+1$:
$3+5+7+...+2n+1+2(n+1)+1=(3+5+7+...+2n+1)+2n+2+1=n(n+2)+2n+3=n^2+2n+2n+3=n^2+n+3n+3=n(n+1)+3(n+1)=(n+1)(n+3)=(n+1)[(n+1)+2]$
That is exactly the given formula if $n$ is changed by $n+1$
