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.