Question

The sum of three numbers in G.P. is $70$; if the two extremes be multiplied each by $4$, and the mean by $5$, the products are in A.P.; find the numbers.

Answer 1

Let $a,ar,a{r}^2$ be three terms of G.P.
Common ratio $=r$
Given :
$a+ar+a{r}^2=70$ .......... $\left(1\right)$
Also, $4a,5ar,4a{r}^2$ are in A.P.
$\therefore 4a+4a{r}^2=2\times 5ar$
$\Rightarrow 2+2r^2=5r$
$\Rightarrow 2r^2-5r+2=0$
$\therefore r=2,\frac{1}{2}$
Putting value of $r=2$ in eqn. $\left(1\right)$, we get
$a(1+2+4)=70$
$\Rightarrow a=10$
Thus, the numbers are $10,20,40$.