Question

Find three numbers which form a geometric progression if their sum is $35$ and the sum of their squares is $525$.

Answer 1

Let the three terms be $a,ar,a{r}^2$

From the given conditions

$a+ar+a{r}^2=\mathrm{35.....}\left(1\right)$

$a^2+a^2{r}^2+a^2{r}^4=\mathrm{525.....}\left(2\right)$

$(1{)}^2-(2)⟹{35}^2-525=2(a^{2}r\left)\right(1+r+r^2)⟹ar=10$

$⟹\frac{10}{r}+10+10r=35⟹\frac{10r^2+10}{r}=25$

$⟹r=2,\frac{1}{2}$

The three numbers are $20,10,5$