Binet's Formula for finding the \(n\)-th Fibonacci number \(F(n)\) is given by:

\[
F(n) = \frac{1}{\sqrt{5}} \left( \phi^n - \hat{\phi}^n \right)
\]
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 find \(F(20)\), we'll apply the formula:

\[
F(20) = \frac{1}{\sqrt{5}} \left( \phi^{20} - \hat{\phi}^{20} \right)
\]

Let's compute the values step by step.

1. \(\phi^{20} \approx 15126.999\)
2. \(\hat{\phi}^{20} \approx -0.000619\)

Now substitute these into the formula:

\[
F(20) = \frac{1}{\sqrt{5}} \left( 15126.999 - (-0.000619) \right)
\]

\[
F(20) = \frac{1}{\sqrt{5}} \times 15127 \approx \frac{15127}{2.236}
\]

\[
F(20) \approx 6765
\]

Thus, \(F(20) = 6765\).

Would you like more details on the calculation process or any further clarifications?

Here are 5 related questions:
1. How does Binet's Formula relate to the Fibonacci sequence?
2. Can Binet's Formula be used for non-integer values of \(n\)?
3. What is the significance of the golden ratio (\(\phi\)) in Fibonacci numbers?
4. How is the Fibonacci sequence related to nature and growth patterns?
5. What are some practical applications of Fibonacci numbers in mathematics and computer science?

**Tip**: When using Binet’s formula, the second term involving \(\hat{\phi}\) becomes very small for larger values of \(n\), making the calculation easier.

Learn how to calculate F(20) using Binet's Formula for Fibonacci numbers. This detailed solution explores algebraic concepts such as the Fibonacci sequence, golden ratio, and the mathematical formula for large Fibonacci numbers. Suitable for students in Grades 10-12.

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