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The formula you provided represents a method to compute the 20th Fibonacci number, \( F_{20} \), using **Binet's formula**. This is expressed as:

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

where \( \phi \) (the golden ratio) is approximately \( 1.618034 \), and \( 1 - \phi \) is approximately \( -0.618034 \). The equation in the image is an approximation for this formula.

The given expression is:

\[
F_{20} = \frac{(1.618034)^{20} - (0.618034)^{20}}{\sqrt{5}}
\]

Let me calculate \( F_{20} \) for you.The value of \( F_{20} \) using the given formula is approximately \( 6765 \), which is the 20th Fibonacci number.

Would you like more details or have any questions about this result?

Here are some related questions you may find interesting:

1. How does Binet's formula approximate Fibonacci numbers?
2. What is the significance of the golden ratio \( \phi \) in the Fibonacci sequence?
3. How accurate is this approximation for larger Fibonacci numbers?
4. Can you compute the 21st Fibonacci number using the same formula?
5. What are some real-world applications of the Fibonacci sequence?

**Tip:** The Fibonacci sequence often appears in nature, such as in the arrangement of leaves or the pattern of a sunflower's seeds!

What is the value of F20 using the provided mathematical expression?

Learn how to compute the 20th Fibonacci number (F20) using Binet's formula, which involves the golden ratio. This mathematical expression helps approximate Fibonacci numbers using algebraic methods. Suitable for high school and advanced students.

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