Question

The sum of three numbers in G.P. is $38$, and their product is $1728$; find them.

Answer 1

Let the three terms of G.P be $\frac{a}{r},a,ar$
Common ratio $=r$
Given that $\frac{a}{r}+a+ar=a(\frac{1}{r}+1+r)=38$ ......(i)
$\frac{a}{r}\times a\times ar=1728$
$\Rightarrow a^3=1728={12}^3$
$\Rightarrow a=12$
Substituting value of $a$ in equation (i), we get
$12(r^2+r+1)=38r$
$\Rightarrow 6r^2-13r+6=0$
$\Rightarrow r=\frac{3}{2}$ or $\frac{2}{3}$
Thus terms are $8,12,18$ or $18,12,8$.