It looks like you're working on a problem involving the Fibonacci sequence and Binet's Formula. The question asks you to solve for $F_{50}$, the 50th Fibonacci number, using Binet's formula.

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

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

This formula calculates the $n$-th Fibonacci number, where $n$ is any integer, like 50 in this case.

Let me calculate $F_{50}$ for you using this formula.Using Binet's formula, the 50th Fibonacci number $F_{50}$ is:

$F_{50} = 12,586,269,025$

Would you like more details on how this calculation works, or do you have any other questions?

Here are 5 related questions for deeper understanding:

1. What is the general form of Binet's formula and how is it derived?
2. Why does Binet's formula work for calculating Fibonacci numbers, even though it involves irrational numbers?
3. Can Binet's formula be applied to negative indices in the Fibonacci sequence?
4. How does the value of the golden ratio $\phi$ relate to the growth of Fibonacci numbers?
5. How would you use Binet's formula to approximate large Fibonacci numbers without a calculator?

Tip: For very large Fibonacci numbers, Binet's formula is efficient but using programming or mathematical software is often easier due to potential rounding errors in manual calculations.