The golden ratio and Binet's formula

Compute the ratio of two consecutive Fibonacci terms:

   F(n+1)/F(n) :  1/1=1   2/1=2   3/2=1.5   5/3=1.667   8/5=1.6   13/8=1.625 ...

These ratios oscillate while closing in on a famous number, the golden ratio, written φ (phi):

   φ = (1 + √5) / 2 ≈ 1.6180339887...

φ is the positive solution of the equation x² = x + 1 — that is, the Fibonacci relation itself, "the next one is the sum of the two before", carried over to powers. The larger n grows, the closer the ratio F(n+1)/F(n) sticks to φ.

Better still: there is a closed-form formula that gives F(n) directly, without unrolling every term. It is Binet's formula:

   F(n) = (φ^n − ψ^n) / √5       with  ψ = (1 − √5) / 2 ≈ −0.618

Try it for n = 10: the computation lands exactly on 55. It is puzzling — a sequence of integers expressed with √5 and irrational powers, yet the result is always an exact integer. The key: the second term ψ^n is tiny, because |ψ| < 1, so ψ^n tends to 0. It is then enough to round:

   F(n) = round( φ^n / √5 )

So the golden ratio is not a curiosity sitting beside Fibonacci: it is its hidden core. Whenever you see the sequence, φ is never far away.