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 numbers.

Answer 1

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