Question

If the sum of the series
$1+\frac{2}{3}+\frac{6}{9}+\frac{10}{27}+\frac{14}{81}+\ldots$ is $2 m-5$, then find the value of $m$.

Answer 1

Let $S=1+\frac{2}{3}+\frac{6}{9}+\frac{10}{27}+\frac{14}{81}+\ldots$
$\Rightarrow S=1+\frac{2}{3}+\frac{6}{3^{2}}+\frac{10}{3^{3}}+\frac{14}{3^{4}}+\ldots$
$\Rightarrow S-1=\frac{2}{3}+\frac{6}{3^{2}}+\frac{10}{3^{3}}+\frac{14}{3^{4}}+\ldots\,\,\,...(i)$
$\Rightarrow \frac{1}{3}(S-1)=\frac{2}{3^{2}}+\frac{6}{3^{3}}+\frac{10}{3^{4}}+\ldots\,\,\,...(ii)$
Subtracting (ii) from (i), we get
$\frac{2}{3}( S -1)=\frac{2}{3}+\frac{4}{3^{2}}+\frac{4}{3^{3}}+\frac{4}{3^{4}}+\ldots$
$\Rightarrow \frac{2}{3}( S -1)=\frac{2}{3}+\frac{\frac{4}{9}}{1-\frac{1}{3}}$
$\Rightarrow \frac{2}{3}( S -1)=\frac{4}{3}$
$\Rightarrow S =3$
$\Rightarrow 2 m-5=3$
$\Rightarrow m=4$