Let $P(n)$ be the given statement i.e. $P(n): 1.2+2.2^2+3.2^2+\ldots \ldots . .+n \cdot 2^n$ $=(n-1) 2^{n+1}+2$
Putting $n=1$, LHS $=1.2=2 ;$ RHS $=0+2=2$
$\therefore P(n)$ is true for $n=1$
Assume that $P(n)$ is true for $n=k$ i.e., $P(k)$ is true i.e., $1.2+2.2^2+3.2^2+\ldots \ldots \ldots . .+k \cdot 2^k$ $=(k-1) 2^{k+1}+2$
We have to prove that $P(k+1)$ is true whenever $P(k)$ is true.
$\therefore \quad P(k+1)$
$=1.2+2 \cdot 2^2+3 \cdot 2^2+\ldots+k \cdot 2^k+(k+1) 2^{k+1}$
$=(k-1) 2^{k+1}+2+(k+1) 2^{k+1}$
$=2^{k+1}[k-1+k+1]+2=2 k \cdot 2^{k+1}+2$
$=k 2^{k+2}+2$
This proves $P(n)$ true for $n=k+1$
Thus $P(k+1)$ is true whenever $P(k)$ is true.
Hence. $P(n)$ is true for all $n \in N$