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

JEE Main · 2016 · 03 Apr 2016 · Offline · Sequences and Series

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:

Answer: (A)

With first term

a

and common difference

d

, the 2nd, 5th and 9th terms are

a + d

a + 4 d

and

a + 8 d

. For a GP the middle term squared equals the product of the other two:

(a + 4 d)^{2} = (a + d) (a + 8 d)

. Expanding,

a^{2} + 8 a d + 16 d^{2} = a^{2} + 9 a d + 8 d^{2}

, which reduces to

8 d^{2} = a d

, so

a = 8 d

(non-constant means

d \neq = 0

). The common ratio is

\frac{a + 4 d}{a + d} = \frac{12 d}{9 d} = \frac{4}{3}

JEE Main · 2016 · 03 Apr 2016 · Offline · Single correct option