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 $AP$.
$\Rightarrow \, d \neq 0$
$\therefore \, d = \frac{a}{8} $
$\Rightarrow a = 8 d$
so, $A.P$ is $8d , 9d ,10 d$ ,....
Common ratio of $G.P. = \frac{t_5}{t_2} = \frac{a+4d}{a+d} = \frac{12d}{9d} = \frac{4}{3}$