Question

The continued product of three numbers in G.P. is $216$, and the sum of the product of them in pairs is $156$; find the numbers.

Answer 1

Let the three terms of G.P be $\frac{a}{r},a,ar$
Common ratio $=r$
Given :
$\frac{a}{r}\times a\times ar=216$
$\Rightarrow a^3=216=6^3$
$\therefore a=6$
Now, $\frac{a}{r}\times a+a\times ar+ar\times \frac{a}{r}=156$ .... [Given]
$\Rightarrow a^2(r+1+\frac{1}{r})=156$
$\Rightarrow 36(r+1+\frac{1}{r})=156$
$3r^2+3r+3=13r$
$3r^2-10r+3=0$
$\Rightarrow r=3,\frac{1}{3}$
Let us take $r=3$
The, the numbers are $\frac{6}{3},6,6.3$ i.e. $2,6,18$