Question

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

Answer 1

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

For n = 1

$1.2={0.2}^o+2$

$2=2$

$\Rightarrow$ P(n) is true for n = 1

Let P(n) is true for n = k, so

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

We have to show that,

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

Now,

$=\{1.2+{2.2}^2+{3.2}^3+........+k{.2}^k\}+(k+1)2^{k+1}$

$=\left[\right(k-1)2^{k+1}+2]+(k+1)2^{k+1}$

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

$=k{2}^{k+2}+2$

$\Rightarrow$ p(n) is true for n = k + 1

$\Rightarrow$ p(n) is true for all n $epsilon$ N by PMI