Binet's Formula is used to find the \(n\)-th Fibonacci number. The formula is given by:

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

Where \(F_n\) is the \(n\)-th Fibonacci number.

To find \(F_{35}\) and then divide by 5, we follow these steps:

1. Calculate \( \frac{1 + \sqrt{5}}{2} \) (denote it as \(\phi\)) and its 35th power.
2. Calculate \( \frac{1 - \sqrt{5}}{2} \) (denote it as \(1 - \phi\)) and its 35th power.
3. Subtract the second value from the first.
4. Divide by \( \sqrt{5} \).
5. Finally, divide the result by 5 to find \( \frac{F_{35}}{5} \).

Let's calculate this using Python for precision.Using Binet's Formula, we find that:

\[
\frac{F_{35}}{5} = 1,845,493
\]

Would you like more details or have any questions about the process?

Here are five related questions to explore further:
1. How is Binet's Formula derived from the Fibonacci sequence?
2. Can Binet's Formula be used to calculate very large Fibonacci numbers efficiently?
3. What is the significance of the golden ratio (\(\phi\)) in Fibonacci numbers?
4. How does Binet's Formula relate to recurrence relations?
5. How would you compute Fibonacci numbers using matrix exponentiation?

**Tip:** Binet’s Formula becomes more accurate for larger Fibonacci numbers because the second term \( \left( \frac{1 - \sqrt{5}}{2} \right)^n \) diminishes rapidly as \( n \) increases.

Discover how to use Binet's Formula to calculate the 35th Fibonacci number (F35) and divide it by 5. This advanced math problem covers key concepts such as the Fibonacci sequence, the golden ratio, and Binet's Theorem. Suitable for students in Grades 11-12 or early college.

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