What is the General Term for the Fibonacci Sequence?

What is the Fibonacci sequence?

The Fibonacci sequence is a sequence of numbers that each number is the sum of the two preceding ones, starting from 0 and 1 (nth term of this sequence is commonly denoted as Fₙ).

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …

F₀ = 0
F₁ = 1 (F₀ = 0 and F₁=1 are definitions)
F₂ = F₁ + F₀ = 1 + 0 = 1
F₃ = F₂ + F₁ = 1 + 1 = 2
F₄ = F₃ + F₂ = 2 + 1 = 3
F₅ = F₄ + F₃ = 3 + 2 = 5
F₆ = F₅ + F₄ = 5 + 3 = 8
……

In this article, we define the 0th term of this sequence: F₀ = 0. But in some cases, F₀ is omitted, and the sequence starts with F₁ = F₂ = 1

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …

In either case, the general term Fₙ will be the same.

How to find the general term of the Fibonacci sequence?

In order to find the general term of the Fibonacci sequence, solve the following recurrence relation.

The meaning of (n ≥ 2) in (*) is that in the case of n < 2, namely n = 0 and n = 1, Fₙ₋₁ and Fₙ₋₂ will not be defined.