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 $|\beta| < 1$.