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 an 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.
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 .
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,
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.
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- G-Fact 18 | Finding nth Fibonacci Number using Golden Ratio
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- Check whether two numbers are in golden ratio
- 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
- Find the number which when added to the given ratio a : b, the ratio changes to c : d
- Sum of Fibonacci numbers at even indexes upto N terms
- Sum of P terms of an AP if Mth and Nth terms are given
- Number of quadruples where the first three terms are in AP and last three terms are in GP
- Minimum Fibonacci terms with sum equal to K
- Sum of nth terms of Modified Fibonacci series made by every pair of two arrays
- Check if a M-th fibonacci number divides N-th fibonacci number
- Check if sum of Fibonacci elements in an Array is a Fibonacci number or not
- Program to find the common ratio of three numbers
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- Section formula (Point that divides a line in given ratio)
- Find if it is possible to get a ratio from given ranges of costs and quantities
- Length and Breadth of rectangle such that ratio of Area to diagonal^2 is maximum
- Program to calculate the profit sharing ratio
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Improved By : AayushChaturvedi