Question

How many terms of series $2+4+8+$ .......... make the sum $1022$.

Answer 1

It is given that the first term of G.P is $a_1=2$ and the second term is $a_2=4$, therefore,

$r=\frac{a_2}{a_1}=\frac{4}{2}=2$

We know that the formula for sum of $n$ terms of G.P is $S_n=\frac{a(r^n-1)}{r-1}$, where $a$ is the first term, $r$ is the common ratio.

In the given geometric series, the first term is $a=2$, the common ratio is $r=2$ and the sum of the terms is $S_n=1022$, therefore,

$S_n=\frac{a(1-r^n)}{1-r}\Rightarrow 1022=\frac{2(2^n-1)}{2-1}\Rightarrow 1022=2(2^n-1)\Rightarrow \frac{1022}{2}=2^n-1\Rightarrow 511=2^n-1\Rightarrow 511+1=2^n\Rightarrow 2^n=512\Rightarrow 2^n=2^9\Rightarrow n=9$

Hence, $9$th term of the series make the sum $1022$.