For the question,
The second term of a GP is half and the sum to infinity is 4. Find the first term and the common ratio of the GP.
For this I reached,
$ar={1 \over 2}, \; {a \over 1-r}=4$
From this I was able to reach several answers for both the value of the first term and the common ratio none of which matching,
$A= 2 \pm \sqrt 2, R = {1 \over 4(2 \pm \sqrt 2)}$ From my manual.