Question

If the sum of three numbers of an infinite GP. is 21 and the sum of their squares is 189 then the common ratio of G.P. is

• A. 2
• B. $\frac{1}{2}$
• C. $\frac{1}{3}$
• D. $\frac{1}{4}$

Answer 1

Answer: (B)

Let the three numbers in G.P. be $a, a r, a r^2$
$a\left(1+r+r^2\right)=21 $ .....(1)
$a^2\left(1+r^2+r^4\right)=189$ .....(2)
$\frac{a^2\left(1+r^2+r^4\right)}{a^2\left(1+r+r^2\right)^2}=\frac{189}{(21)^2}$
$\frac{r^2-r+1}{r^2+r+1}=\frac{3}{7} $
$ 4 r^2-10 r+4=0$
$\Rightarrow 2 r^2-5 r+2=0 $
$ 2 r^2-4 r-r+2=0$
$2 r(r-2)-1(r-2)=0$
$\therefore r=2 \text { or } r=\frac{1}{2}$
(but $r=2$ is rejected)