Question

Three numbers are in G.P. whose sum is 70. If the extremes be each multiplied by 4 and the means by 5, they will be in A.P. Find the largest of those numbers divided by smallest of those numbers ?

Answer 1

Let the numbers be $a,ar,a{r}^2$
$\therefore Sum=a(1+r+r^2)=70...\left(1\right)$ Also
$4a,5ar,4a{r}^2$
$\Rightarrow \frac{5}{2}r=1+r^2...\left(2\right)$
or $2r^2-5r+2=0or(r-2)(2r-1)=0$
$\therefore r=2$ or $1/2$
Hence from (1), $a=10,40$
Ergo the numbers are $10,20,40$ or $40,20,10.$