Question

The sum of an infinite geometric series is $512$ and sum of first $n$ terms is $510$ ($n>1$) . If the inverse of its common ratio is an integer then which of this following can not be the second term of the series?

• A. $30$
• B. $96$
• C. $256$
• D. $128$

Answer 1

The correct option is C $256$

Let $a$ be the 1st term of the series and $r$ be thier common ratio.
$\Rightarrow S_{\infty }=\frac{a}{1-r}=512$
and
$S_n=\frac{a(1-r^n)}{1-r}=510$
$\Rightarrow \frac{S_{\infty }}{S_n}=\frac{256}{255}=\frac{1}{1-r^n}\Rightarrow r^n=\frac{1}{256}$
Since, $\frac{1}{r}$ is given as an integer and $n$ is always an integer therefore,
${\left(\frac{1}{r}\right)}^n=256$
$\Rightarrow \frac{1}{r}$ can be $2,4,16$ for $n=8,4,2$
$\Rightarrow a=256,384,480\Rightarrow a_2=128,96,30$