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.

Explore how to calculate the 50th Fibonacci number using the Binet formula. This solution provides a detailed breakdown of the Fibonacci sequence, Binet's formula, and the role of the golden ratio. Suitable for students in Grades 10-12.

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