Exercise about geometric progession

Question

I have an statement that says:

Find 3 numbers in P.G, if this numbers add up $12$ and its product is $-216$

I tried this:

The $r = \frac{\pm\sqrt{5} - 3}{2}$

And when i replace the $r$, it not give $12$ in the addition

Answer 1

We have \begin{eqnarray*} r=\frac{-3 \pm \sqrt{5}}{2} \\ \end{eqnarray*} which gives \begin{eqnarray*} a=3(3 \pm \sqrt{5}) \\ \end{eqnarray*} and so ($ar=-6$) \begin{eqnarray*} a+ar+ar^2 =3(3 \pm \sqrt{5})-6-3(-3 \pm \sqrt{5})=12 \\ \end{eqnarray*} as required.