# If the second, fourth and ninth term of an arithmetic progression are in geometric progression, then what is the common ratio of the GP?

Dec 4, 2017

Please refer to the Discussion in The Explanation.

#### Explanation:

Let the AP be, $a , a + d , a + 2 d , \ldots , a + \left(n - 1\right) d , \ldots \left(n \in \mathbb{N} .\right) .$

Then, by what is given, $a + d , \left(a + 3 d\right) \mathmr{and} \left(a + 8 d\right)$ are in GP.

$\therefore {\left(a + 3 d\right)}^{2} = \left(a + d\right) \left(a + 8 d\right) .$

$\therefore {a}^{2} + 6 a d + 9 {d}^{2} = {a}^{2} + 9 a d + 8 {d}^{2} , \mathmr{and} ,$

${d}^{2} - 3 a d = 0 , i . e . ,$

$d \left(d - 3 a\right) = 0.$

$\therefore d = 0 , \mathmr{and} , d = 3 a .$

If $d = 0 ,$ then, the GP becomes the constant sequence

$a , a , a , \ldots a , \ldots$ for which the common ratio is $1. \left(a \ne 0\right)$

In case, $d = 3 a ,$ then the GP is $4 a , 10 a , 25 a$ for which the

common ratio is $2.5 a , \mathmr{and} , \frac{5}{2} a , \left(a \ne 0.\right)$