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 $a,ar,a{r}^2$
$a\left(1+r+r^2\right)=21$ .....(1)
$a^2\left(1+r^2+r^4\right)=189$ .....(2)
$\frac{a^2\left(1+r^2+r^4\right)}{a^2{\left(1+r+r^2\right)}^2}=\frac{189}{(21{)}^2}$
$\frac{r^2-r+1}{r^2+r+1}=\frac{3}{7}$
$4r^2-10r+4=0$
$\Rightarrow 2r^2-5r+2=0$
$2r^2-4r-r+2=0$
$2r(r-2)-1(r-2)=0$
$\therefore r=2\text{ or }r=\frac{1}{2}$
(but $r=2$ is rejected)