The Fibonacci Numbers have been of great interest since they were first seen as a solution to an algebra word problem published in 1202. The Fibonacci Numbers have many interesting properties which have many applications. For this the focus will be on finding a formula to find the nth Fibonacci number in the sequence.

The sequence is set up so that –

The pattern for generating new terms relies on the previous two terms and is written as

For any integer value of n 0 or greater.

This is called the renewal equation. It uses information from the current and previous steps to generate future steps. It is also a second order difference equation.

Because it is a difference equation this may be rewritten as follows:

Since this is a difference equation that is linear, a solution can be made from the linear superposition of the known solutions.

This is for 0 or greater integer values of n=0,1,2,3…

This is a system of irrational numbers that always answers with an integer.

Also $$\lambda_1\approx 1.6180339887…$$ is the value known as the Golden Section or Golden Ratio.

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