Question

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

Answer 1

Let the three numbers in G.P be $\frac{a}{r},a,ar$
then $\frac{a}{r}+a+ar=70\cdots \left(1\right)$
and $\frac{4a}{r},5a,4ar$ are in A.P.
$\therefore 10a=\frac{4a}{r}+4ar$
$\Rightarrow \frac{10a}{4}=\frac{a}{r}+ar$
$\Rightarrow \frac{5a}{2}=70-a$ from $\left(1\right)$
$\Rightarrow 5a=140-2a$
$\Rightarrow 7a=140$
$\therefore a=20$
from $\left(1\right)$, $\frac{20}{r}+20+20r=70$
$\Rightarrow \frac{20}{r}+20r=50$
$\Rightarrow 2+2r^2=5r$
$\Rightarrow 2r^2-5r+2=0$
$\Rightarrow (r-2)(2r-1)=0$
$\therefore r=2$ or $\frac{1}{2}$
$\therefore$ The three numbers are $10,20,40$ or $40,20,10$