Fibonacci & φ

Rabbits, spirals and the golden ratio

An interactive lesson with 6 steps, about 5 minutes: you predict, drag and solve, with instant feedback on every answer. It runs in your browser and needs JavaScript.

What you’ll learn

• Fibonacci: $F_{n+1} = F_n + F_{n-1}$.
• Ratios of neighbouring terms approach $\varphi = \frac{1+\sqrt5}{2} \approx 1.618$.
• $\varphi^2 = \varphi + 1$.

Steps

1. Next in line
2. Ratios of neighbours
3. The golden ratio
4. A neat property
5. Cassini’s identity
6. Sum of Fibonaccis