# Prove the following by using the principle of mathematical induction for all $n \in N$  $1.2+2.2^2+3.2^2+...+n.2^n=(n-1)2^{n+1}+2$

Let the given statement be $P(n)$, i.e.,
$P(n) : 1.2+2.2^2+3.2^2+...+n.2^n=(n-1)2^{n+1}+2$
For $n = 1$, we have
$P(1) : 1.2 = 2 =(1-1)2^{1+1}+2=0+2=2$, which is true.
Let $P(k)$ be true for some positive integer $k$, i.e.,
$1.2+2.2^2+3.2^2+...+k.2^k=(k-1)2^{k+1}+2$------------(i)
We shall now prove that $P(k+1)$ is true.
Consider
$\{1.2+2.2^2+3.2^3+...+k.2^k \}+(k+1).2^{k+1}$
$=(k-1)2^{k+1}+2+(k+1)2^{k+1}$
$= 2^{k+1} \{ (k-1)+(k+1) \}+2$
$2^{k+1}.2k+2$
$k.2^{(k+1)+1}+2$
$\{ (k+1)-1 \} 2^{(k+1)+1}+2$
Thus, $P(k+1)$ is true whenever $P(k)$ is true.
Hence, by the principle of mathematical induction, statement $P(n)$ is true for all natural numbers i.e., $n$.