Binet's formula for the Fibonacci sequence is given by:

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

where:
- \(\alpha = \frac{1 + \sqrt{5}}{2}\) (the golden ratio),
- \(\beta = \frac{1 - \sqrt{5}}{2}\).

To find \( \frac{F_{35}}{5} \), let's proceed step by step:

### Step 1: Calculate \( \alpha^{35} \) and \( \beta^{35} \).
- \(\alpha = \frac{1 + \sqrt{5}}{2} \approx 1.6180339887\)
- \(\beta = \frac{1 - \sqrt{5}}{2} \approx -0.6180339887\)

### Step 2: Apply Binet's formula for \( F_{35} \).
\[
F_{35} = \frac{1}{\sqrt{5}} \left( \alpha^{35} - \beta^{35} \right)
\]

Let me calculate this for you.Using Binet's formula, \( \frac{F_{35}}{5} \approx 1845493 \).

Would you like a detailed explanation of the steps, or do you have any questions?

Here are 5 related questions for further exploration:
1. How does Binet's formula approximate Fibonacci numbers for large \( n \)?
2. What is the significance of the golden ratio in the Fibonacci sequence?
3. Can Binet’s formula be applied to non-integer Fibonacci indices?
4. How would you derive Binet's formula from the recurrence relation of Fibonacci numbers?
5. How does \( F_n \) grow asymptotically for large \( n \)?

**Tip**: The ratio of successive Fibonacci numbers converges to the golden ratio as \( n \) increases.

Discover how to calculate F35/5 using Binet's formula. This solution covers key mathematical concepts such as the Fibonacci sequence, the golden ratio, and Binet's formula. Ideal for high school students in Grades 10-12.

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