The formula provided for $F_n$ appears to be a variant of the closed-form expression for Fibonacci numbers, known as Binet's formula. However, the formula as written seems unclear due to the notation. The correct Binet's formula is:

$F_n = \frac{1}{\sqrt{5}} \left[ \left(\frac{1 + \sqrt{5}}{2}\right)^n - \left(\frac{1 - \sqrt{5}}{2}\right)^n \right]$

We are tasked with finding $F_{13}$, so let's calculate it using the correct formula.

Step 1: Identify the constants

• $\phi_1 = \frac{1 + \sqrt{5}}{2}$
• $\phi_2 = \frac{1 - \sqrt{5}}{2}$

Step 2: Calculate $\phi_1^{13}$ and $\phi_2^{13}$

Let's begin by calculating these powers.

Step 3: Apply the formula for $F_{13}$

$F_{13} = \frac{1}{\sqrt{5}} \left(\phi_1^{13} - \phi_2^{13}\right)$

Now, let’s compute it.

$\phi_1 = \frac{1 + \sqrt{5}}{2} \approx 1.6180339887$ $\phi_2 = \frac{1 - \sqrt{5}}{2} \approx -0.6180339887$

So,

$\phi_1^{13} \approx 1.6180339887^{13} \approx 521.025$ $\phi_2^{13} \approx (-0.6180339887)^{13} \approx -0.000025$

Now,

$F_{13} = \frac{1}{\sqrt{5}} \left(521.025 - (-0.000025)\right) \approx \frac{521.025 + 0.000025}{2.236067977} \approx \frac{521.025}{2.236067977} \approx 233$

Therefore,

$F_{13} = 233$

Would you like more details on the calculation process, or do you have any other questions?

Here are five related questions:

1. How is Binet's formula derived from the Fibonacci sequence?
2. Can Binet's formula be applied to negative indices of the Fibonacci sequence?
3. What is the significance of the golden ratio ($\phi$) in Fibonacci numbers?
4. How would you calculate $F_{13}$ using a recursive method instead of Binet's formula?
5. What are some real-world applications of Fibonacci numbers?

Tip: When dealing with Fibonacci numbers, Binet's formula offers a direct method to compute any Fibonacci number without the need for recursion.