Why Fibonacci fails both sequence tests
The Fibonacci sequence runs 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, .... Its consecutive differences are 0, 1, 1, 2, 3, 5, 8, 13, 21 — not constant, so it isn't arithmetic. Its consecutive ratios are 1, 2, 1.5, 1.667, 1.6, 1.625, 1.615, 1.619, 1.618 — also not constant, so it isn't geometric either. See the companion article on telling arithmetic and geometric sequences apart for what those two tests actually check.
But the ratios aren't random — they converge
Look at that list of ratios again: 1, 2, 1.5, 1.667, 1.6, 1.625, 1.615, 1.619, 1.618. After the first few terms, they stop swinging as widely and start narrowing in on one number. Continuing further makes this clearer — F(11)/F(10) = 89/55 ≈ 1.618182, F(12)/F(11) = 144/89 ≈ 1.617978, F(13)/F(12) = 233/144 ≈ 1.618056, F(14)/F(13) = 377/233 ≈ 1.618026, F(15)/F(14) = 610/377 ≈ 1.618037. Each ratio overshoots and undershoots the same target by a smaller amount than the last.
The number they're converging to
That target is φ (phi), the golden ratio, equal to (1 + √5) / 2 ≈ 1.618034. It's an irrational number — Fibonacci ratios never land on it exactly at any finite term, since every Fibonacci ratio is a ratio of two whole numbers and φ isn't one — but they get arbitrarily close as the term number grows, alternating between slightly above and slightly below it.
Why this happens
Each Fibonacci term is the sum of the two before it, so the ratio of consecutive terms is F(n)/F(n−1) = (F(n−1) + F(n−2)) / F(n−1) = 1 + F(n−2)/F(n−1) — meaning each ratio is 1 plus the reciprocal of the previous ratio. Once the ratio stabilizes near a value x, that relationship becomes x = 1 + 1/x, which rearranges to x² − x − 1 = 0 — and the positive solution to that equation is exactly φ. The convergence isn't a coincidence of the particular starting values 1, 1; any sequence that adds the previous two terms to get the next one converges to the same ratio.
A reasonable caveat
The golden ratio's appearance in Fibonacci numbers is genuine and provable, unlike some of the more decorative golden-ratio claims made about art, architecture, and the human body that circulate online — many of those are approximate at best or don't hold up to scrutiny. This article only claims what's mathematically demonstrable: the ratio of consecutive Fibonacci terms converges to φ as the term number grows.