Geometric Sequences and the Common Ratio (F 703)

An arithmetic sequence adds the same amount each step. A geometric sequence multiplies by the same amount each step. That multiplying is why geometric growth outruns arithmetic growth so fast: adding 2 repeatedly gives you 2, 4, 6, 8, 10 while multiplying by 2 repeatedly gives you 2, 4, 8, 16, 32. The gap between terms itself grows in a geometric sequence, because each new gap is the old gap times the ratio again. After only ten or so steps a doubling sequence has pulled miles ahead of an adding sequence, even though both started the same way.

To find the common ratio, pick any term and divide it by the term right before it. If that division gives the same number no matter which neighboring pair you check, the sequence is geometric and that number is r. Once you know the first term a(1) and the ratio r, you can reach any term without listing every step in between: a(n) = a(1)·r^n-1, since getting from term 1 to term n takes exactly n - 1 multiplications by r.

This sits right next to arithmetic sequences in the same unit, and the two are easy to mix up when a sequence just looks like it is growing fast. Fast growth alone does not make something geometric. A sequence like a(n) = 5n² climbs by bigger and bigger jumps, but the ratio of consecutive terms keeps changing as n grows, so it fails the one test that matters: a constant ratio between neighbors.