Question

Find smallest of the three numbers in G.P. having sum $19$ and product $216.$

Answer 1

Let the three numbers be $\frac{a}{r},a,ar$
so $a[\frac{1}{r}+1+r]=19$ (i)
and $a^3=216$ $\Rightarrow$ $a=6$
so from (i) $6r^2-13r+6=0$ $\Rightarrow$ $r=\frac{3}{2},\frac{2}{3}$
Hence the three numbers are $4,6,9.$