Question

Sum the following series
$1+\frac{2}{3}+\frac{2}{2^2}+\frac{4}{2^3}+....$ to $n$ terms.

Answer 1

We have,

$S=\frac{1}{2}+\frac{2}{2^2}+\frac{4}{2^3}+\frac{8}{2^4}\dots$ to $n$ terms

Any $rth$ term of the series is given by,

$T_r=\frac{2^{n-1}}{2^n}=\frac{1}{2}$

$\therefore S=\frac{1}{2}+\frac{1}{2}+\frac{1}{2}\dots \dots n$ terms

$\Rightarrow S=\frac{n}{2}$