A Formula for Fibonacci

Binet’s formula

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

• Try $a_n = x^n$ to turn a linear recurrence into a polynomial equation.
• Binet: $F_n = (\varphi^n - \psi^n)/\sqrt5$.
• $F_n$ is the integer nearest to $\varphi^n/\sqrt5$.

Steps

1. Guess an exponential
2. Check it
3. Ratios approach φ
4. Same method, new rule
5. The magic ratio
6. The 20th Fibonacci number