Question

Find the sum of n terms of the series $\frac{19}{\mathrm{1.2.3}}\cdot \frac{1}{4}+\frac{28}{\mathrm{2.3.4}}\cdot \frac{1}{8}+\frac{39}{\mathrm{3.4.5}}\cdot \frac{1}{16}+\frac{52}{\mathrm{4.5.6}}\cdot \frac{1}{32}+....$

Answer 1

$\Rightarrow t_n=\frac{n^2+6n+12}{n(n+1)(n+2)}\cdot \frac{1}{2^{n+1}}$

$=\frac{n+3}{n(n+1)}\cdot \frac{1}{2^n}-\frac{n+4}{(n+1)(n+2)}\cdot \frac{1}{2^{n+1}}$

$S_n=1-\frac{n+4}{(n+1)(n+2)}\cdot \frac{1}{2^{n+1}}$