Solve the febonacci squence f35 golden ratio

To solve for the 35th Fibonacci number (\(F_{35}\)) using the golden ratio, we can use Binet's formula. Binet's formula is an explicit formula for the \(n\)th Fibonacci number and is given by:

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

where:
- \(F_n\) is the \(n\)th Fibonacci number,
- \(\phi = \frac{1 + \sqrt{5}}{2}\) is the golden ratio (approximately 1.61803),
- \(1 - \phi = \frac{1 - \sqrt{5}}{2}\) (approximately -0.61803).

For \(n = 35\), the formula becomes:

\[
F_{35} = \frac{\phi^{35} - (1-\phi)^{35}}{\sqrt{5}}
\]

Let's calculate this step by step.

1. Calculate \(\phi^{35}\).
2. Calculate \((1-\phi)^{35}\).
3. Subtract the second result from the first.
4. Divide by \(\sqrt{5}\).

I'll perform this calculation now.The 35th Fibonacci number (\(F_{35}\)) calculated using the golden ratio is approximately 9,227,465.

Would you like

Learn how to solve the 35th Fibonacci number using the golden ratio and Binet's formula. Explore step-by-step calculations and understand the application of Binet's formula in Fibonacci sequences.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation