For every positive integer $n$,${2^n} < n!$ when

Prove the following by using the principle of mathematical induction for all $n \in N:$
$a+(a+d)+(a+2d)+\ldots+(a+(n-1)d) = \frac{n}{2}[2a+(n-1)d]$

Using mathematical induction,the numbers $a_n$ are defined by:
$a_0 = 1, a_{n+1} = 3n^2 + n + a_n, (n \geq 0)$.
Then,$a_n$ is equal to:

For all natural numbers $n$,$3(5^{2n+1}) + 2^{3n+1}$ is divisible by:

Prove that the following is true for all $n \in N$ using the principle of mathematical induction:
$n(n+1)(n+5)$ is a multiple of $3.$

Prove the following by using the principle of mathematical induction for all $n \in N$:
$\left(1+\frac{3}{1}\right)\left(1+\frac{5}{4}\right)\left(1+\frac{7}{9}\right) \dots \left(1+\frac{2n+1}{n^{2}}\right)=(n+1)^{2}$