Question

Find three numbers in G.P, whose product is $216$ and the sum of their products is pairs is $156$.

Answer 1

Let 3 numbers in GP

$\begin{array}{cc}\frac{a}{r}\cdot a\cdot ar& \\ now,& \\ \frac{a}{r}\cdot a\cdot ar=216& \\ a^3=216& \\ a=6& \\ \frac{a}{r}\cdot a/ar+ar\cdot \frac{a}{r}=156& \\ \frac{36}{r}+36r+36=156& \\ \frac{36}{r}+36r=156-36& \\ \frac{36}{r}+36r=120& \\ 36\left[\frac{1}{r}+r\right]=120& \\ 36\left[1+\frac{r^2}{r}\right]=120& \\ 36r^2-120r+36=0& \\ {90}^2-30r+9=0& \\ 3\left(3r^2-10r+3\right)=0& \\ 3r^2-10r+3=0& \\ 3r^2-9r-r+3=0& \\ 3r\left(r-3\right)-1\left(r-3\right)=0& \\ \left(3r-1\right)\left(r-3\right)=0& \\ 3r-1=0& r-3=0\\ r=\frac{1}{3}& r=3\\ ifr=\frac{1}{3}\&a=6& \\ then,a_1=\frac{a}{r}& \\ =\frac{6}{\frac{1}{3}}& \\ =18& \\ a_2=a=6& \\ a_3=ar& \\ =6\times \frac{1}{3}& \\ =2& \\ ifr=3\&a=6& \\ then,& \\ a_1=\frac{a}{r}& \\ =\frac{6}{3}& \\ =2& \\ a_2=a=6& \\ a_3=ar=18& \end{array}$