Mathematics - Sequence & Series Question with Solution | TestHub

Given, $S_n=\frac{1}{1^3}+\frac{1+2}{1^3+2^3}+\frac{1+2+3}{1^3+2^3+3^3}+\dots +\frac{1+2+\dots +n}{1^3+2^3+\dots +n^3}$

$T_n=\frac{\frac{n\left(n+1\right)}{2}}{{\left(\frac{n\left(n+1\right)}{2}\right)}^2}$

$T_n=\frac{2}{n\left(n+1\right)}=2\left(\frac{1}{n}-\frac{1}{n+1}\right)$

$S_n=2\sum _{n=1}^n\left(\frac{1}{n}-\frac{1}{n+1}\right)$

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

Given, $100S_n=n\Rightarrow 100\times \frac{2n}{n+1}=n$

$n+1=200$

$n=199$