1.2 + 2.2

Let P(n) be the given statement.

Now,

$P(n)=1.2+2.2^{2}+3.2^{3}+\ldots+n .2^{n}=(n-1) 2^{n+1}+2$

Step 1:

$P(1)=1.2=2=(1-1) 2^{1+1}+2$

Thus, $P(1)$ is true.

Step 2 :

Let $P(m)$ be true.

Then,

$1.2+2.2^{2}+\ldots+m .2^{m}=(m-1) 2^{m+1}+2$

To prove: $P(m+1)$ is true.

That is,

$1.2+2.2^{2}+\ldots+(m+1) 2^{m+1}=m .2^{m+2}+2$

Now,

$P(m)=1.2+2.2^{2}+\ldots+m .2^{m}=(m-1) 2^{m+1}+2$

$\Rightarrow 1.2+2.2^{2}+\ldots+m .2^{m}+(m+1) \cdot 2^{m+1}=(m-1) 2^{m+1}+2+(m+1) \cdot 2^{m+1}$ $\left[\right.$ Adding $(m+1) \cdot 2^{m+1}$ to both sides $]$

$\Rightarrow P(m+1)=2 m \cdot 2^{m+1}+2=m \cdot 2^{m+2}+2$

Thus, $P(m+1)$ is true.

$B y$ the $p$ rinciple of $m$ athematical $i$ nduction, $P(n)$ is true for all $n \in N$.