Question

The second, the first, and the third term of an arithmetic progression, whose common difference is nonzero, form a geometric progression in that order. Find its common ratio.

Answer 1

Let the 3 terms of the AP be $(a-d),a,(a+d)$

Terms of the GP: $a,(a-d),(a+d)$ in that order.

In a GP, terms next to each other have the same ratio.

So, $\frac{(a-d)}{a}=\frac{(a+d)}{(a-d)}$

$(a-d{)}^2=a(a+d)$

$d^2-2ad=ad$

$d^2-3ad=0$

$d(d-3a)=0$

We know that d is not 0 from the question. So $d=3a$

Common ratio$=\frac{(a-d)}{a}=\frac{(a-3a)}{a}=-2$