Question

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

Answer 1

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