Question

If the $2^{nd},5^{th}$ and $9^{th}$ terms of a non-constant $A.P.$ are in $G.P.$, then the common ratio of this $G.P.$ is :

• A. $\frac{8}{5}$
• B. $\frac{4}{3}$
• C. 1
• D. $\frac{7}{4}$

Answer 1

Answer: (B)

$t_2=a+d$
$t_5=a+4d$
$t_9=a+8d$
Given $t_2,t_5,t_9$ are in $G.P$.
$(a+4d{)}^2=(a+d)(A+8d)$
$a^2+16d^2+8ad=a^2+8d^2+9ad$
$8d^2-ad=0$
$d(8d-a)=0$
As given non - constant $AP$.
$\Rightarrow d\ne 0$
$\therefore d=\frac{a}{8}$
$\Rightarrow a=8d$
so, $A.P$ is $8d,9d,10d$ ,....
Common ratio of $G.P.=\frac{t_5}{t_2}=\frac{a+4d}{a+d}=\frac{12d}{9d}=\frac{4}{3}$