Find the sum of first $n$ terms and the sum of first $5$ terms of the geometric
series $1+\frac{2}{3}+\frac{4}{9}+\ldots$
Find the sum of first $n$ terms and the sum of first $5$ terms of the geometric
series $1+\frac{2}{3}+\frac{4}{9}+\ldots$
Here $a=1$ and $r=\frac{2}{3} .$ Therefore
$S_{n}=\frac{a\left(1-r^{n}\right)}{1-r}=\frac{\left[1-\left(\frac{2}{3}\right)^{n}\right]}{1-\frac{2}{3}}=3\left[1-\left(\frac{2}{3}\right)^{n}\right]$
In particular, $S_{5}=3\left[1-\left(\frac{2}{3}\right)^{5}\right]=3 \times \frac{211}{243}=\frac{211}{81}$
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