Question

An infinite geometric series has sum 50. A new series, obtained by squaring each term of the original series, has 10 times the sum of the original series. The common ratio of the original series is $\frac{ m }{ n }$ where $m$, $n$ are relatively prime integers. Find the value of $( m + n )$.

Answer 1

Let $a, a r, a r^2, \ldots \ldots . .$. be an infinite G.P.
So, $\frac{ a }{1- r }=50$....(1)
Also, $\frac{ a ^2}{1- r ^2}=10 \cdot 50 \Rightarrow \frac{ a }{1+ r }=10$ .....(2) [Using equation (1)]
$\therefore \frac{\text { Equation (1) }}{\text { Equation }(2)}=\frac{1+r}{1-r}=5 \Rightarrow 1+r=5-5 r \Rightarrow 6 r=4 \Rightarrow r=\frac{2}{3} $
$\therefore m+n=5 $