Question

The product of three consecutive terms of a geometric progression is $216$ and the sum of their products taken in pairs is $156$. Find the terms of the progression.

Answer 1

Solution:

Let the terms be $\frac{a}{r},a,ar$

Given:

Product $=216$

or, $\frac{a}{r}\times a\times ar=216$

or, $a^3=216$

or, $a=6$

And

$(\frac{a}{r}\times a)+(a\times ar)+(ar\times \frac{a}{r})=156$

or, $\frac{a^2}{r}+a^{2}r+a^2=156$

or, $\frac{36}{r}+36r+36=156$

or, $\frac{36}{r}+36r=156-36$

or, $\frac{36}{r}+36r=120$

or, $36r^2-120r+36=0$

or, $3r^2-10r+3=0$

or, $3r^2-9r-r+3=0$

or, $3r(r-3)-1(r-3)=0$

or, $(r-3)(3r-1)=0$

or, $r=3$ or $r=\frac{1}{3}$

The terms of progression are

$2,6,18$ or $18,6,2$