Question

Find the sum of first $n$ terms and the sum of first $5$ terms of the geometric series $1+\frac{2}{3}+\frac{4}{9}+...$

Answer 1

Given,

$a=1$

$r=\frac{\frac{2}{3}}{1}=\frac{2}{3}$

Formula,

$S_n=\frac{a(1-r^n)}{1-r}$

$=\frac{1(1-{\left(\frac{2}{3}\right)}^n)}{1-\frac{2}{3}}$

$=\frac{(1-{\left(\frac{2}{3}\right)}^n)}{\frac{1}{3}}$

$\therefore S_n=3[1-{\left(\frac{2}{3}\right)}^n]$

$S_5=3[1-{\left(\frac{2}{3}\right)}^5]$

$=3[1-\left(\frac{32}{243}\right)]$

$=3\left[\frac{211}{243}\right]$

$=\frac{211}{81}$