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

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

It is given that 2nd, 5th and 9th terms of A.P. are in G.P., hence,

$\frac{a+4d}{a+d}=\frac{a+8d}{a+4d}$

where, $a$ is the first term and $d$ is the common difference of A.P.

$\Rightarrow (a+4d{)}^2=(a+8d\left)\right(a+d)$

$\Rightarrow a^2+16d^2+8ad=a^2+8d^2+9ad$

$\Rightarrow 8d^2=ad$

$\Rightarrow a=8d$ ...($d\ne 0$ since A.P is not constant)

$\therefore$Common ratio of G.P.$=$$\frac{a+4d}{a+d}$$=$$\frac{12d}{9d}$$=$$\frac{4}{3}$