Fibonacci Sequence

The Fibonacci sequence is a series of numbers in which each number is the sum of the two preceding ones, usually starting with 0 and 1. The sequence starts with 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so on.

Formula for the Fibonacci Sequence

The formula to find the nth number in the Fibonacci sequence is:

F_n = F_n-1 + F_n-2

Example

To find the 7th number in the Fibonacci sequence:

F₇ = F₆ + F₅ = 8 + 5 = 13

Properties of the Fibonacci Sequence

1. The ratio of consecutive Fibonacci numbers approaches the golden ratio, which is approximately 1.618.
2. The Fibonacci sequence is closely related to the golden ratio, which appears in many natural objects and has been used in art and architecture for centuries.

Applications of the Fibonacci Sequence

The Fibonacci sequence has applications in various fields such as mathematics, computer science, and finance. It is also found in nature, for example in the arrangement of leaves on a stem, the fruitlets of a pineapple, the flowering of artichokes, and the uncurling of a fern.

Study Guide

To study the Fibonacci sequence, it's important to:

1. Understand the recursive nature of the sequence, where each number is the sum of the two preceding ones.
2. Practice using the formula to find specific numbers in the sequence.
3. Explore the connections between the Fibonacci sequence and the golden ratio.
4. Research real-world applications of the Fibonacci sequence in various fields.

Additionally, practicing with Fibonacci sequence problems and exploring its properties will help in mastering this topic.