Question

Find the numbers in G.P., whose sum is $52$ and the sum of whose products in pairs is $624$.

Answer 1

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

$\frac{a}{r}+a+ar=52$

$a(\frac{a}{r}+1+r)=52$

$\Rightarrow a^2{(\frac{a}{r}+1+r)}^2=2704$...(1)

$\frac{a}{r}\cdot a+a\cdot ar+ar\cdot \frac{a}{r}=624$

$a^2(\frac{a}{r}+1+r)=624$...(2)

$\left(2\right)÷\left(1\right)$ gives

$\frac{a}{r}+1+r=\frac{13}{3}$

$\Rightarrow 3r^2-10r+3=0$

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

when $r=3,a=12$

$\therefore \frac{a}{r},a,ar=\frac{12}{3},12,\left(12\right)\left(3\right)=4,12,36$