The Binet formula is used to find the $n$-th term in the Fibonacci sequence without the need to calculate all the previous terms. The Fibonacci sequence is defined by the recurrence relation:

$F_n = F_{n-1} + F_{n-2}$ with initial conditions $F_0 = 0$ and $F_1 = 1$.

The Binet formula 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 is an exact expression for the $n$-th Fibonacci number. Let's calculate $F_{50}$ using this formula.

We'll substitute $n = 50$ into the formula to compute the 50th Fibonacci number.

$F_{50} = \frac{1}{\sqrt{5}} \left( \left(\frac{1 + \sqrt{5}}{2}\right)^{50} - \left(\frac{1 - \sqrt{5}}{2}\right)^{50} \right)$

Let me calculate that for you.The 50th Fibonacci number, $F_{50}$, using Binet's formula is approximately $12,586,269,025$.

Would you like more details or have any questions about the solution?

Here are some related questions to consider:

1. How can the Binet formula be derived?
2. What are the properties of the golden ratio $\phi$?
3. How accurate is the Binet formula for large values of $n$?
4. Can you apply the Binet formula to negative Fibonacci indices?
5. What are some applications of Fibonacci numbers in real-world problems?

Tip: The second term in Binet's formula involving $\psi$ (negative golden ratio) quickly becomes negligible for large $n$, simplifying calculations.