Question

Find the three numbers in G.P. whose sum is $52$ and sum of whose product in pairs is $624$.

• A. $2,6,18$
• B. $4,16,64$
• C. $4,12,36$
• D. $2,8,32$

Answer 1

The correct option is C $4,12,36$

Solution:- (C) 4, 12, 36

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

Given that sum is 52.

$\therefore a+ar+a{r}^2=52$

$\Rightarrow a(1+r+r^2)=52⟶\left(1\right)$

Sum of product of numbers in pairs is 624.

$\therefore a.ar+ar.a{r}^2+a{r}^2.a=624$

$\Rightarrow a^{2}r(1+r^2+r)=624$

From ${eq}^n\left(1\right)$, we have

$ar=\frac{624}{52}=12$

$\Rightarrow a=\frac{12}{r}$

Substituting the value of a, in ${eq}^n\left(1\right)$, we have

$\frac{12}{r}(1+r+r^2)=52$

$\Rightarrow 3+3r+3r^2=13r$

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

$\Rightarrow 3r^2-9r-r+3=0$

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

$\Rightarrow r=3\text{or}\frac{1}{3}$

For $r=3$,

$a=\frac{12}{r}=4$

Required numbers are 4, 12, 36.

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

$a=\frac{12}{r}=36$

Required numbers are $36,12,4$.