Find three numbers in G.P. whose sum is 65 and whose product is 3375.

Solution:

Let the terms of the the given G.P. be $\frac{a}{r}, a$ and $a r$.

Then, product of the G.P. = 3375

= a3 = 3375

= a = 15

Similarly, sum of the G.P. = 65

$\Rightarrow \frac{a}{r}+a+a r=65$

Substituting the value of a

$\frac{15}{r}+15+15 r=65$

$\Rightarrow 15 r^{2}+15 r+15=65 r$

$\Rightarrow 15 r^{2}-50 r+15=0$

$\Rightarrow 5\left(3 r^{2}-10 r+3\right)=0$

$\Rightarrow 3 r^{2}-10 r+3=0$

$\Rightarrow(3 r-1)(r-3)=0$

$\Rightarrow r=\frac{1}{3}, 3$

Hence, the G.P. for $a=15$ and $r=\frac{1}{3}$ is $45,15,5$.

And, the G.P. for a = 15 and r = 3 is 5, 15, 45.