If $P\left(n\right)$ is the statement $n(n+1)(n+2)$ is divisible by $12$, show that $P\left(3\right)$ and $P( 4)$ are true but not $P\left(5\right)$.

.Use mathematical induction to prove that 

$\left(1+\frac{3}{1}\right)·\left(1+\frac{5}{4}\right)\cdots \left(1+\frac{2n+1}{n^2}\right)=(n+1{)}^2$,for all $n\in N$.

Prove using the principle of mathematical induction for all $n\in \mathrm{N}$ that

$1·3+3·5+5·7+\cdots +(2n-1)(2n+1)=\frac{n\left(4n^2+6n-1\right)}{3}$