Binet's formula is a closed-form expression for the Fibonacci sequence \( F_n \), given by:

\[
F_n = \frac{1}{\sqrt{5}} \left( \alpha^n - \beta^n \right)
\]
where:
- \( \alpha = \frac{1 + \sqrt{5}}{2} \) (the golden ratio),
- \( \beta = \frac{1 - \sqrt{5}}{2} \),
- \( F_n \) is the \( n \)-th Fibonacci number.

We are tasked with calculating \( F_{30} + F_{35} \). To do so, let's compute each Fibonacci number using Binet's formula.

### Step 1: Compute \( F_{30} \)

\[
F_{30} = \frac{1}{\sqrt{5}} \left( \alpha^{30} - \beta^{30} \right)
\]

### Step 2: Compute \( F_{35} \)

\[
F_{35} = \frac{1}{\sqrt{5}} \left( \alpha^{35} - \beta^{35} \right)
\]

### Step 3: Find \( F_{30} + F_{35} \)

\[
F_{30} + F_{35} = \frac{1}{\sqrt{5}} \left( \alpha^{30} - \beta^{30} \right) + \frac{1}{\sqrt{5}} \left( \alpha^{35} - \beta^{35} \right)
\]

This simplifies to:

\[
F_{30} + F_{35} = \frac{1}{\sqrt{5}} \left( \alpha^{30} + \alpha^{35} - \beta^{30} - \beta^{35} \right)
\]

Let's now calculate the numerical values.

I'll calculate this for you now.The sum of \( F_{30} \) and \( F_{35} \) is approximately \( 10,059,505 \).

Would you like more details on the calculation or have any questions? Here are 5 related questions to explore further:

1. What are the first few Fibonacci numbers?
2. How is the golden ratio related to Fibonacci numbers?
3. Can Binet's formula be used for non-integer indices?
4. How accurate is Binet's formula for large Fibonacci numbers?
5. What is the relationship between Fibonacci numbers and the Lucas sequence?

**Tip**: Binet's formula is exact but for large \( n \), the term \( \beta^n \) becomes very small and negligible due to the fact that \( |\beta| < 1 \).

Learn how to use Binet's formula to find F30 + F35 in the Fibonacci sequence. This detailed explanation covers the use of the golden ratio and provides a step-by-step solution for computing Fibonacci numbers. Suitable for high school students in Grades 10-12.

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