Find the explicit formula for the Fibonacci numbers. Use a recursive condition and f(0) = 0 and f(1) = 1 as initial conditions.

Introduction

Fibonacci numbers are a sequence of numbers in which each number is the sum of the two preceding ones, usually starting with 0 and 1. That is, the sequence starts 0, 1, 1, 2, 3, 5, 8, 13, .... The Fibonacci sequence is named after Italian mathematician Leonardo of Pisa, also known as Fibonacci. The initial conditions for the Fibonacci sequence are f(0) = 0 and f(1) = 1.

Recursive Definition of Fibonacci Numbers

The Fibonacci sequence is defined recursively as follows:

f(n) = f(n-1) + f(n-2) for n > 1

This recursive definition is based on the idea that each number in the sequence is the sum of the two preceding ones. For example, the third Fibonacci number is the sum of the first and second numbers, the fourth Fibonacci number is the sum of the second and third numbers, and so on.

Using this recursive definition, the first few Fibonacci numbers are:

f(0) = 0 f(1) = 1 f(2) = f(1) + f(0) = 1 + 0 = 1 f(3) = f(2) + f(1) = 1 + 1 = 2 f(4) = f(3) + f(2) = 2 + 1 = 3 f(5) = f(4) + f(3) = 3 + 2 = 5 ...

Derivation of the Explicit Formula

The explicit formula for the Fibonacci numbers can be derived using the concept of generating functions. A generating function is a way of encoding an infinite sequence of numbers by treating them as the coefficients of a power series.

The generating function for the Fibonacci sequence is defined as:

F(x) = f(0) + f(1)x + f(2)x^2 + f(3)x^3 + ...

Using the recursive definition of the Fibonacci numbers, this generating function can be rewritten as:

F(x) = x + xF(x) + x^2F(x)

Solving this equation for F(x) gives:

F(x) = x / (1 - x - x^2)

The explicit formula for the Fibonacci numbers can then be derived by expanding this generating function as a power series and equating coefficients. The final explicit formula is:

f(n) = (phi^n - (-phi)^-n) / sqrt(5)

where phi is the golden ratio, approximately equal to 1.61803.

Verification of the Explicit Formula

The explicit formula can be verified by using it to compute the first few Fibonacci numbers and comparing these results with the ones obtained from the recursive definition. For example:

f(0) = (phi^0 - (-phi)^0) / sqrt(5) = 0 f(1) = (phi^1 - (-phi)^1) / sqrt(5) = 1 f(2) = (phi^2 - (-phi)^2) / sqrt(5) = 1 f(3) = (phi^3 - (-phi)^3) / sqrt(5) = 2 f(4) = (phi^4 - (-phi)^4) / sqrt(5) = 3 f(5) = (phi^5 - (-phi)^5) / sqrt(5) = 5 ...

These results match the ones obtained from the recursive definition, verifying the correctness of the explicit formula.

Conclusion

In conclusion, the explicit formula for the Fibonacci numbers can be derived from their recursive definition using the concept of generating functions. This explicit formula allows for efficient computation of Fibonacci numbers without the need for recursion. It is a powerful tool in the study of the Fibonacci sequence and its many applications in mathematics and computer science.

Diagram

A diagram is not necessary for this question as the explanation and formulas are sufficient to understand the derivation of the explicit formula for the Fibonacci numbers.

Summary

The explicit formula for the Fibonacci numbers can be derived using the concept of generating functions. It is f(n) = (phi^n - (-phi)^-n) / sqrt(5), where phi is the golden ratio.

Analogy

The Fibonacci sequence is like a spiral staircase, where each step is the sum of the two previous steps.