Question

The product of five positive number in G.P is $32$ and ratio of the greatest number to the smallest number is $81:1$, find the numbers.

Answer 1

Let the numbers be,

$\frac{a}{r^2},\frac{a}{r},a,ar,a{r}^2$

Given,

$\frac{a}{r^2}\times \frac{a}{r}\times a\times ar\times a{r}^2=32$

$a^5=32$

$\therefore a=2$

$\frac{a{r}^2}{\frac{a}{r^2}}=\frac{81}{1}$

$r^4=81$

$\therefore r=3$

Therefore the numbers are, $\frac{2}{3^2},\frac{2}{3},2,2\left(3\right),2(3{)}^2$

$\Rightarrow \frac{2}{9},\frac{2}{3},2,6,18$