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:
1. 1
2. 7/4
3. 8/5
4. 4/3

Answer 1

Correct Answer - Option 4 : 4/3

Concept:

The general form of n^th term of A.P is given by 

T_n = a + (n - 1)d, where a - first term, d - common difference.

For non constant A.P, r ≠ 0;

If three consecutive terms (a, b, c) are in G.P, then 

b² = ac;

Calculation:

Given 2nd,5th and 9th terms of a non-constant A.P. are in G.P;

⇒ T₂ = a + d, T₅ = a + 4d, T₉ = a + 8d;

G.P ⇒ (a + 4d)² = (a +d)(a + 8d)

⇒ a² + 16d² + 8ad = a² + 9ad + 8d²;

⇒ 8d² = ad ⇒ a = 8d;

The common ratio of G.P will be T₅/T₂ or T₉/T₅;

⇒ r = (a + 4d)/(a + d) = 12d/9d = 4/3