Find the common ratio of the progression.

Question

A geometric progression has 625 as the first term. The product of its first 3 terms is equal to the product of its first 6 terms. Find the common ratio of the progression.

Answer 1

Write the terms of the progression as $a_n = 625 r^n$. This gives that our first term, $a_0$, is $625 = 5^4$; call this $a$. The product of the first three terms is hence $a \cdot ar \cdot ar^2 = a^3r^3$, and similarly, we have the product of the first six terms is $a^6r^{15}$. Thus either $r=0$ or $$a^3 r^3 = a^6 r^{15},$$ giving $a^3r^{12} = 1$, or $5^{12} r^{12} = 1$.

So $r = \pm\frac 15$ or $0$.

edit as Yves correctly points out, $r=0$ isn't actually allowed in the definition of a GP, so unless otherwise specified, the answer is $\pm\frac 15$.