To calculate $F_{50}$, the 50th Fibonacci number, using Binet's Formula, we use the following expression:

$F_n = \frac{{\phi^n - (1 - \phi)^n}}{\sqrt{5}}$

Where:

• $\phi = \frac{{1 + \sqrt{5}}}{2}$ is the golden ratio.
• $1 - \phi = \frac{{1 - \sqrt{5}}}{2}$.
• $n$ is the position in the Fibonacci sequence, in this case $n = 50$.

Let's calculate $F_{50}$ using this formula. I'll start by plugging in these values and performing the necessary computations.Using Binet's formula, the 50th Fibonacci number $F_{50}$ is:

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

Would you like more details or have any questions on how this result was derived?

Here are five related questions to explore further:

1. What is Binet's formula used for in mathematics?
2. How does the Fibonacci sequence appear in nature and mathematics?
3. Can we calculate Fibonacci numbers for large values of $n$ efficiently?
4. What is the golden ratio, and why is it important in Binet's formula?
5. How does the recursive definition of Fibonacci numbers differ from Binet's formula?

Tip: Binet’s formula provides an exact Fibonacci number for any given $n$, but for very large $n$, numerical methods like matrix exponentiation might be more efficient to avoid rounding errors.