Calculate the 23rd Fibonacci Number using the Binet Formula 

The Fibonacci numbers are defined by the recurrence relation:

\[
F_n = F_{n-1} + F_{n-2}
\]
with \( F_0 = 0 \) and \( F_1 = 1 \). 

The **Binet formula** for the \(n\)-th Fibonacci number is:

\[
F_n = \frac{1}{\sqrt{5}} \left( \phi^n - \hat{\phi}^n \right)
\]

where:
- \( \phi = \frac{1 + \sqrt{5}}{2} \) (the golden ratio)
- \( \hat{\phi} = \frac{1 - \sqrt{5}}{2} \) (the conjugate of the golden ratio)

Now, to compute the 23rd Fibonacci number \(F_{23}\), we substitute \(n = 23\) into the formula:

\[
F_{23} = \frac{1}{\sqrt{5}} \left( \phi^{23} - \hat{\phi}^{23} \right)
\]

Let's calculate this.The 23rd Fibonacci number, calculated using the Binet formula, is \( F_{23} = 28,657 \).

Would you like more details on how the Binet formula works, or do you have any other questions?

Here are 5 related questions to explore:
1. How is the Binet formula derived?
2. What are the limitations of using the Binet formula for very large Fibonacci numbers?
3. What is the 24th Fibonacci number?
4. Can the Fibonacci sequence be generalized to other sequences?
5. How is the Fibonacci sequence connected to the golden ratio?

**Tip:** The Binet formula is exact, but rounding errors can occur for very large Fibonacci numbers due to floating-point arithmetic limitations.

Calculate the 23rd Fibonacci Number using the Binet Formula

Learn how to calculate the 23rd Fibonacci number using Binet's formula. This method applies the golden ratio and explores the relationship between Fibonacci numbers and number theory. Suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation