Question

Two distinct, real, infinite geometric series each have a sum of 1 and have the same second term. The third term of one of the series is $1 / 8$. If the second term of both the series can be written in the form $\frac{\sqrt{ m }- n }{ p }$ where $m, n$ and $p$ are positive integers and $m$ is not divisible by the square of any prime, find the value of $100 m+10 n+p$

Answer 1

Let $a_1, r_1$ be the first term and the common ratio of the first series.
hence $\frac{a_1}{1-r_1}=1 \Rightarrow a_1+r_1=1$
also $a_1 r_1^2=\frac{1}{8} \Rightarrow a_1=\frac{1}{8 r_1^2}$
substituting in (1)
$\frac{1}{8 r_1^2}+r_1=1 \Rightarrow 8 r_1^3-8 r_1^2+1=0$
$4 r_1^2\left(2 r_1-1\right)-2 r_1\left(2 r_1-1\right)-1\left(2 r_1-1\right)=0$
$\left(2 r_1-1\right)\left(4 r_1^2-2 r_1-1\right)=0$
hence $r_1=\frac{1}{2} $ or $ r_1=\frac{2 \pm \sqrt{4+16}}{8}=\frac{1 \pm \sqrt{5}}{4}$
$\therefore r_1=\frac{1}{2} \text { or } r_1=\frac{1+\sqrt{5}}{4} \text { or } r_1=\frac{1-\sqrt{5}}{4}$
corresponding values of $1^{\text {st }}$ term are,
$\left. a _1=\frac{1}{2} \text { or } a _1=\frac{3-\sqrt{5}}{4} \text { or } a _1=\frac{3+\sqrt{5}}{4} \text { [using } a _1+ r _1=1\right] $
$\text { now } 2^{\text {nd }} \text { term } a_1 r_1=\frac{1}{4} \text { or } \frac{\sqrt{5}-1}{8} \text { or } \frac{-\sqrt{5}-1}{8} $
only the $2^{\text {nd }}$ is of the given form
hence $m =5 ; n =1 ; p =8$
$\therefore 100 m +10 n + p =518 $