Question

For an infinite G.P, the common ratio is selected from $1,\frac{1}{2},\frac{1}{4},\frac{1}{8}$. The sum of the series converges to $1/7$. If $1/2^a$ is the first term of such a G.P., find a such that 'a' is an integer.

Answer 1

Let common ratio is $\frac{1}{2^b}$
and $S_{\infty }=\frac{a}{1-r}=\frac{\frac{1}{2^a}}{1-\frac{1}{2^b}}=\frac{1}{7}$
$\Rightarrow b=3$ & $a=b$
$\Rightarrow b=3$ & $a=b$
Hence $a=3$