If the [2^nd,5^th,and,9^th] terms of a non-constant A.P.

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 :

Accepted Answer

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.

Solutions · 1