# Deriving the expression of Fibonacci Numbers in terms of golden ratio

**Prerequisites:** Generating Functions, Fibonacci Numbers, Methods to find Fibonacci numbers.

The method of using Generating Functions to solve the famous and useful Fibonacci Numbers‘ recurrence has been discussed in this post.

The **Generating Function** is a powerful tool for solving a wide variety of mathematical problems, including counting problems. It is a formal power series. For example, in counting problems, we are often interested in finding the number of objects of size . In such a case, we define a power series, which, in simple terms is an infinite term polynomial where the coefficient of the degree term is the term of the sequence. This helps us to find many interesting and important results. It should be noted that in the use of generating functions, we generally use the coefficients in the generating function power series, we rarely use the variable in the series. In this post too, we shall do the same. The ordinary generating function of some a_{n} is:

Fibonacci Numbers are one of the fundamental sequences in mathematics, and numerous ways have been discovered to find out the higher order terms of this sequence. This post discusses one such method.

Let’s first define a generating function for the Fibonacci Numbers, and then the function will be simplified to get a recurrence. Using this, expand the simplification and break it into partial fractions, and then use two standard power series, and later combine them both to arrive at surprising result for the term of the Fibonacci Series.

Let us define the generating function as

,

where is the ith Fibonacci Number.

Since,

.

.

Rearranging them we get,

.

Taking the common terms,

Simplifying it further, the below function is obtained.

.

Solving for , we get:

.

We get the below formula by above operations:

,

where, and .

Thus,

Also note that,

.

Thus, keeping this relation in above expression, we get,

.

Now the right hand side of the above expression can be separated into partial fractions,

.

Using Expansion on the two fractions,

.

Similarly,

.

Thus,

.

Thus,

.

Now,

,

and,

Using the above two facts, it can be safely concluded that the value of

, rounded to the nearest integer.

Finding n-th Fibonacci number using Golden ratio is one of the applications of this formula.

## Recommended Posts:

- Find nth Fibonacci number using Golden ratio
- G-Fact 18 | Finding nth Fibonacci Number using Golden Ratio
- Sum of Fibonacci numbers at even indexes upto N terms
- Ratio of mth and nth terms of an A. P. with given ratio of sums
- Find n terms of Fibonacci type series with given first two terms
- Sum of two numbers if the original ratio and new ratio obtained by adding a given number to each number is given
- Minimum Fibonacci terms with sum equal to K
- Sum of nth terms of Modified Fibonacci series made by every pair of two arrays
- Program to find the common ratio of three numbers
- Find the number which when added to the given ratio a : b, the ratio changes to c : d
- Find a permutation of 2N numbers such that the result of given expression is exactly 2K
- Sum of Fibonacci Numbers
- Non Fibonacci Numbers
- GCD and Fibonacci Numbers
- Even Fibonacci Numbers Sum

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