Mathematics - Sequence & Series Question with Solution | TestHub

$1+\frac{3}{2}+\frac{7}{4}+\frac{15}{8}+\frac{31}{16}+\dots$

$=\left(2-1\right)+\left(2-\frac{1}{2}\right)+\left(2-\frac{1}{4}\right)+\left(2-\frac{1}{8}\right)+\left(2-\frac{1}{16}\right)+\dots \mathrm{u}\mathrm{p}\mathrm{t}\mathrm{o}\mathrm{ }20\mathrm{ }\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{m}\mathrm{s}$

$=40-\left(1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\dots \mathrm{u}\mathrm{p}\mathrm{t}\mathrm{o} 20 \mathrm{t}\mathrm{e}\mathrm{r}\mathrm{m}\mathrm{s}\right)$

$=40-\left(\frac{1-{\left(\frac{1}{2}\right)}^{20}}{1-\frac{1}{2}}\right)=40-2+\frac{1}{2^{19}}=38+\frac{1}{2^{19}}$