Prove the following by using the principle of mathematical induction for all n ∈ N: 1.2 + 2.22 + 3.22

Solution:

Let the given statement be P(n), i.e.,

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

For $n=1$, we have

$P(1): 1.2=2=(1-1) 2^{1+1}+2=0+2=2$, which is true.

Let $\mathrm{P}(k)$ be true for some positive integer $k$, i.e.,

$1.2+2.2^{2}+3.2^{2}+\ldots+k .2^{k}=(k-1) 2^{k+1}+2 \ldots$ (i)

We shall now prove that $P(k+1)$ is true.

Consider

$\left\{1.2+2.2^{2}+3.2^{3}+\ldots+k .2^{k}\right\}+(k+1) \cdot 2^{k+1}$

$=(k-1) 2^{k+1}+2+(k+1) 2^{k+1}$

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

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

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

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

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.