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 goes: 0, 1, 1, 2, 3, 5, 8, 13, 21, and so on. The sequence was first described in the year 1202 by Italian mathematician Fibonacci in his book Liber Abaci.

Formula

The formula to find the nth term of the Fibonacci sequence is:

F_n = F_n-1 + F_n-2

Where F_n is the nth term, F_n-1 is the (n-1)th term, and F_n-2 is the (n-2)th term.

Properties of the Fibonacci Sequence

1. The sequence starts with 0 and 1.
2. Each subsequent number in the sequence is the sum of the previous two.
3. The ratio of two consecutive Fibonacci numbers approaches the golden ratio, approximately 1.618, as the sequence progresses.
4. The Fibonacci sequence has connections to various natural phenomena, such as the arrangement of leaves on a stem, the breeding patterns of rabbits, and the shape of galaxies.

Study Guide

To understand the Fibonacci sequence, follow these steps:

1. Memorize the first few terms of the sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...
2. Understand the formula F_n = F_n-1 + F_n-2 and how it generates the sequence.
3. Practice finding the nth term of the sequence using the formula.
4. Explore the connection between the Fibonacci sequence and the golden ratio.
5. Research real-world examples of the Fibonacci sequence in nature and art.

By following this study guide, you will gain a solid understanding of the Fibonacci sequence and its applications.