Question

Find the sum of n terms of the series $\frac{7}{\mathrm{1.2.3}}-\frac{17}{\mathrm{2.3.4}}+\frac{31}{\mathrm{3.4.5}}-\frac{49}{\mathrm{4.5.6}}+\frac{71}{\mathrm{5.6.7}}-.....$

Answer 1

$\Rightarrow$ The general term of the series;-

$7,17,31,49,71,.....$ is

$7+10(n-1)+2(n-1)(n-2)$$=2n^2+4n-1$

$\therefore u_n=\frac{2n^2+4n-1}{n(n+1)(n+2)}=\frac{1}{2}(\frac{1}{n}+\frac{2}{n+1}+\frac{1}{n+2})$

$2S_n=(\frac{1}{1}+\frac{2}{2}+\frac{1}{3})-(\frac{1}{2}+\frac{2}{3}+\frac{1}{4})+(\frac{1}{3}+\frac{2}{4}+\frac{1}{5})-....+{(-1)}^{n-1}(\frac{1}{n}+\frac{2}{n+1}+\frac{1}{n+2})$

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

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