Difficulty finding common ratio and first term.

Question

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.

Answer 1

Here is an outline of the steps:

From $\frac{a}{1-r}=4$, we get $a=4-4r$.

Plug this in to $ar = \frac12$ to get $(4-4r)r=\frac12$.

Rearrange a bit to get $8r^2-8r+1=0$.

Apply quadratic formula to get $r=\frac{2\pm\sqrt{2}}{4}$.