Question

Let $r$ be the positive real zero of $P(x)=9 x^5+7 x^2-9$. If the sum $S=r^4+2 r^9+3 r^{14}+4 r^{19}+$..... $\infty$ can be expressed as the rational number $\left(\frac{ a }{ b }\right)$ in the lowest term, then find the sum of digits in $( a + b )$.

Answer 1

$=\frac{r^4}{1-r^5} $
$S=\frac{r^4}{\left(1-r^5\right)^2}$
Using $1-r^5=\frac{7 r^2}{9}$
$S =\frac{81}{49}=\frac{ a }{ b } \Rightarrow( a + b )=130 $
$\therefore \text { Sum of digits }=4 $