The Fibonacci Sequence

The Fibonacci sequence is a series of numbers in which each number after the first two is the sum of the two preceding ones. It is named after Leonardo Fibonacci, an Italian mathematician who introduced the sequence to Western mathematics.

The definition of the Fibonacci sequence is:

F_n = F_{n-1} + F_{n-2} for n \geq 2

But for the first two terms:

F_0 = 0

F_1 = 1

So we start off with 0 and 1, then add the previous two terms from there.

Therefore, the first few terms of the Fibonacci sequence are:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, \dots

Properties of the Fibonacci Sequence

One interesting property of the Fibonacci sequence is that the sum of the squares of any two consecutive terms is also a term in the sequence. For example:

  • 1^2 + 1^2 = 2
  • 1^2 + 2^2 = 5
  • 5^2 + 8^2 = 25 + 64 = 89

As 2, 5 and 89 are terms in the Fibonacci sequence.

Applications of the Fibonacci Sequence

The Fibonacci sequence can be found in various aspects of nature, art, and architecture. Some examples include the arrangement of leaves on a stem, the growth pattern of seashells, and the construction of the pyramids.

You’ve used 0 of your 10 free revision notes for the month

Sign up to get unlimited access to revision notes, quizzes, audio lessons and more

Sign up