common ratio

The common ratio is the fixed number you multiply by to step from one term to the next in a geometric sequence. If each year your savings grow by half again, the common ratio is 1.5.

You find it by dividing any term by the one before it: r = a_2 / a_1 = a_3 / a_2, and so on. If |r| > 1 the terms grow without bound; if |r| < 1 they shrink toward 0; a negative r makes the signs flip back and forth from term to term.

The size of r decides the long-run behaviour and, for an infinite geometric series, whether the sum settles to a finite value. As with arithmetic sequences, “common” means the same ratio holds for every adjacent pair — if it does not, the sequence is not geometric.