Question

Let the sum of the first $n$ terms of the series : $1+2(1+\frac{1}{n})+3{(1+\frac{1}{n})}^2+4{(1+\frac{1}{n})}^3+......$ be $n^{k-1}$.Find $k$ ?

Answer 1

$S=1+2(1+\frac{1}{n})+3{(1+\frac{1}{n})}^2+4{(1+\frac{1}{n})}^3+...n{(1+\frac{1}{n})}^{n-1}$ ...(1)
$(1+\frac{1}{n})S=(1+\frac{1}{n})+2{(1+\frac{1}{n})}^2+3{(1+\frac{1}{n})}^3+...(n-1){(1+\frac{1}{n})}^{n-1}+n{(1+\frac{1}{n})}^n$ ...(2)
Subtracting (2) from (1), we get
$(1-(1+\frac{1}{n}))S=1+(1+\frac{1}{n})+{(1+\frac{1}{n})}^2+{(1+\frac{1}{n})}^3+...+{(1+\frac{1}{n})}^{n-1}-n{(1+\frac{1}{n})}^n$
$=1+(1+\frac{1}{n})\left(\frac{{(1+\frac{1}{n})}^n-1}{(1+\frac{1}{n})-1}\right)-n{(1+\frac{1}{n})}^n$
$\left(\frac{-1}{n}\right)S=1+(1+\frac{1}{n})\left(\frac{{(1+\frac{1}{n})}^n-1}{\frac{1}{n}}\right)-n{(1+\frac{1}{n})}^n$
$=1+n(1+\frac{1}{n})({(1+\frac{1}{n})}^n-1)-{n(1+\frac{1}{n})}^n$
$=1+n{(1+\frac{1}{n})}^{n+1}-n(1+\frac{1}{n})-n{(1+\frac{1}{n})}^n$
$\Rightarrow S=n^2\Rightarrow S=n^{3-1}\therefore k=3$