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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 :

JEE · Math · previous-year question

  1. A.1
  2. B.$${7 \over 4}$$
  3. C.$${8 \over 5}$$
  4. D.$${4 \over 3}$$correct

Answer

D. $${4 \over 3}$$

Explanation

The terms of an Arithmetic Progression (A.P.) are given by $a$, $a + d$, $a + 2d$, ..., where $a$ is the first term and $d$ is the common difference. Given that the 2nd, 5th and 9th terms of an A.P. are in Geometric Progression (G.P.), we can denote them as follows : 2nd term = $a + d$ 5th term = $a + 4d$ 9th term = $a + 8d$ For three numbers to be in G.P., the square of the middle term must be equal to the product of the other two terms. So, $(a + 4d)^2 = (a + d)(a + 8d)$ Expanding and simplifying : $a^2 + 8ad + 16d^2 = a^2 + 9ad + 8d^2$ $8ad + 16d^2 = a^2 + 9ad + 8d^2$ $8ad - 9ad = 8d^2 - 16d^2$ $-ad = -8d^2$ $a = 8d$ The common ratio of the G.P. is the ratio of the 5th term to the 2nd term, or $(a + 4d) / (a + d)$. Substituting $a = 8d$ gives : $(8d + 4d) / (8d + d) = 12d / 9d = 4 / 3$ So, the common ratio of the G.P. is $4 / 3$. The answer is option D.

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