Mathematics - Sequence & Series Question with Solution | TestHub

Let $S=\frac{1}{3}+\frac{5}{9}+\frac{19}{27}+\cdots$ upto $100$ terms

$S=\frac{3-2}{3^1}+\frac{3^2-2^2}{3^2}+\frac{3^3-2^3}{3^3}+....$ upto $100$ terms

So, $T_n=1-{\left(\frac{2}{3}\right)}^n$$\Rightarrow S_{100}=100-\left[\frac{\frac{2}{3}\left[1-{\left(\frac{2}{3}\right)}^{100}\right]}{1-\frac{2}{3}}\right]$

$=98+2{\left(\frac{2}{3}\right)}^{100}$  $\Rightarrow \left[S_{100}\right]=98$