Using mathematical induction to P.T statement  $2+3.2+{4.2}^2+\dots \dots +$ upto $n$ terms $=n\cdot 2^n,\mathrm{\forall }n\in N$

2,3,4........ are in A.P

$\Rightarrow t_n=a+(n-1)d$

$=2+(n-1)1=2+n-1=n+1$

$1,2,2^2$      ....... are in G.P

$t_n=a{r}^{n-1}={1.2}^{n-1}=2^{n-1}$

$\therefore n^{\text{th }}$ term $=(n+1)2^{n-1}$

$S\left(n\right)=2+3.2+4\cdot 2^2+\dots \dots +(n+1)2^{n-1}$

$=n\cdot 2^n$

Let $S\left(n\right)$ be the given statement.

put $n=1$ 

L.H.S $=(1+1)2^{1-1}={2.2}^0=2$

R.H.S $=1\cdot 2^1=2$

$\Rightarrow$L.H.S = R.H.S

$\therefore S\left(1\right)$ is true

Assume that $S\left(k\right)$ is true for some $k\in N$

$2+3\cdot 2+4\cdot 2^2+\dots \dots +(k+1)2^{k-1}=k\cdot 2^k$

adding $(k+1{)}^{\text{th }}$ term on both sides

$2+3.2+{4.2}^2+\dots .+(k+1)2^{k-1}+(k+2)2^{\mathrm{k}}$

$=k\cdot 2^k+(k+2)2^k$

$=2^k(k+k+2)$

$=2^k(2k+2)=2^k\cdot 2(k+1)=(k+1)2^{k+1}$

$\therefore S(k+1)$ is true

By the principle of finite mathematical induction
$S\left(n\right)$ is true $\mathrm{\forall }n\in N$