• The Fibonacci Sequence
  • Definition
  • First Few Terms
  • Closed-Form Expression (Binet’s Formula)
  • Golden Ratio Connection
  • Sum Formulas

§The Fibonacci Sequence

§Definition

The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding ones. It starts with 0 and 1.

$$F_0 = 0, \quad F_1 = 1$$

$$F_n = F_{n-1} + F_{n-2} \quad \text{for } n > 1$$

§First Few Terms

$$0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, \ldots$$

§Closed-Form Expression (Binet’s Formula)

The nth Fibonacci number can be calculated directly using:

$$F_n = \frac{\varphi^n - \psi^n}{\sqrt{5}}$$

Where:

• \(\varphi = \frac{1 + \sqrt{5}}{2} \approx 1.618\) (the golden ratio)
• \(\psi = \frac{1 - \sqrt{5}}{2} \approx -0.618\)

§Golden Ratio Connection

As \(n\) approaches infinity, the ratio of consecutive Fibonacci numbers converges to the golden ratio:

$$\lim_{n \to \infty} \frac{F_{n+1}}{F_n} = \varphi = \frac{1 + \sqrt{5}}{2}$$

§Sum Formulas

Sum of first \(n\) Fibonacci numbers:

$$\sum_{i=0}^{n} F_i = F_{n+2} - 1$$

Sum of squares:

$$\sum_{i=0}^{n} F_i^2 = F_n \cdot F_{n+1}$$