Question

Find three numbers in G.P. whose sum is $19$, and whose product is $216$.

Answer 1

Denote the numbers by $\frac{a}{r},a,ar$; then $\frac{a}{r}\times a\times ar=216$; hence $a=6$, and the numbers are $\frac{6}{r},6,6r$.
$\therefore \frac{6}{r}+6+6r=19$;
$\therefore 6-13r+6r^2=0$;
hence $r=\frac{3}{2}$ or $\frac{2}{3}$.
Thus the numbers are $4,6,9$.