Question

Find the sum of n terms of the series $\frac{1}{1.2}+\frac{5}{2.3}\cdot 2+\frac{10}{3.4}\cdot 2^2+\frac{17}{4.5}\cdot 2^3+....$

Answer 1

$\Rightarrow t_n=\frac{n^2+1}{n(n+1)}\cdot 2^{n-1}$

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

$=\frac{2n}{n+1}-\frac{n-1}{n}$

$t_n=\frac{2^n\cdot n}{n+1}-\frac{2^{n-1}(n-1)}{n}$

$t_{n-1}=\frac{2^{n-1}(n-1)}{n}-\frac{2^{n-2}(n-2)}{n-1}$

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$t_2=\frac{4\cdot 2}{3}-\frac{2\cdot 1}{2}$

$t_1=\frac{2\cdot 1}{2}-0$

$S_n=\frac{2^n\cdot n}{n+1}$