Question

If the 2nd, 5th and 9th terms of a non-constant AP are in GP, then the common ratio of this GP is

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

Answer 1

The correct option is A $\frac{4}{3}$

Let a be the first term and d be the common difference. Then, we have
$a+d,a+4d,a+8d$ in GP
i.e., $(a+4d{)}^2=(a+d\left)\right(a+8d)$
$\Rightarrow a^2+16d^2+8ad=a^2+8ad+ad+8d^2$
$\Rightarrow 8d^2=ad$
$\Rightarrow 8d=a$ $[\because d\ne 0]$
Now, common ratio,
$r=\frac{a+4d}{a+d}=\frac{8d+4d}{8d+d}=\frac{12d}{9d}=\frac{4}{3}$