How many terms of $G.P.$ $3,3^{2}, 3^{3}$... are needed to give the sum $120 ?$
How many terms of $G.P.$ $3,3^{2}, 3^{3}$... are needed to give the sum $120 ?$
The given $G.P.$ is $3,3^{2}, 3^{3} \ldots$
Let $n$ terms of this $G.P.$ be required to obtain in the sum as $120 .$
$S_{n}=\frac{a\left(1-r^{n}\right)}{1-r}$
Here, $a=3$ and $r=3$
$\therefore S_{n}=120=\frac{3\left(3^{n}-1\right)}{3-1}$
$\Rightarrow 120=\frac{3\left(3^{n}-1\right)}{2}$
$\Rightarrow \frac{120 \times 2}{3}=3^{n}-1$
$\Rightarrow 3^{n}-1=80$
$\Rightarrow 3^{n}=81$
$\Rightarrow 3^{n}=3^{4}$
$\therefore n=4$
Thus, four terms of the given $G.P.$ are required to obtain the sum as $120 .$
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