Ex.l By method of induction, prove that. 1.3 +2.5 +3.7 + ... +'n (2n+1) = (n+1)(4n+5 for all n e N

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

Prove by the principle of mathematical induction that $1+2+3+...+n<\frac{(2n+1{)}^2}{8}$, for all $n\in N.$

Or

Using the principle of mathematical induction, prove that

$1.3+{2.3}^2+{3.3}^3+...+n{.3}^n=\frac{(2n-1)3^{n+1}+3}{4}.$ for all $n\in N.$

Using Principle of Mathematical Induction, prove that
$$1.3 + 2.3 ^ { 2 } + 3.3 ^ { 3 } + \ldots . n .3 ^ { n } = \dfrac { ( 2 n - 1 ) 3 ^ { n + 1 } + 3 } { 4 } ,$$ where $$n \in \mathrm { N }$$

Using principle of mathematical induction, prove that ${41}^n-{14}^n$ is a multiple of 27.

Or
Prove by the principle of mathematical induction $n<2^n$ for all $nϵN$.

Prove by method of induction, for all $n\in$
$3+7+11+...$ to $n$ terms $=n(2n+1)$