1.3: A geometric series with a surd sum

A geometric series has first term $\displaystyle a$ and common ratio $\displaystyle r$, where $\displaystyle r>0$.

Given that the 3rd term of the series is $\displaystyle 5$ and that the 5th term of the series is $\displaystyle \frac{5}{2}$,

(a) find

(i) the exact value of $\displaystyle r$,

(ii) the value of $\displaystyle a$.

(b) Find the sum to infinity of this series. Give your answer in the form

where $\displaystyle p$ and $\displaystyle q$ are integers.