To solve for the \( F_{13} \)th term in Binet's formula, we will use Binet's formula for the Fibonacci sequence. Binet's formula is given by:

\[
F_n = \frac{\varphi^n - \psi^n}{\sqrt{5}}
\]

Where:

- \( \varphi = \frac{1 + \sqrt{5}}{2} \) (the golden ratio)
- \( \psi = \frac{1 - \sqrt{5}}{2} \)
- \( n \) is the term number in the Fibonacci sequence.

For \( F_{13} \), we substitute \( n = 13 \) into Binet's formula:

\[
F_{13} = \frac{\varphi^{13} - \psi^{13}}{\sqrt{5}}
\]

Let’s calculate it step by step.The 13th Fibonacci number \( F_{13} \) calculated using Binet's formula is 233.

Would you like more details on the calculation, or do you have any other questions?

Here are some related questions:

1. What is Binet's formula, and why does it work for Fibonacci numbers?
2. How do you derive Binet's formula?
3. Can Binet's formula be used for non-integer values of \( n \)?
4. What is the significance of the golden ratio \( \varphi \) in Fibonacci numbers?
5. How do Fibonacci numbers appear in nature?
6. Can Binet's formula be used to compute very large Fibonacci numbers?
7. What is the error when approximating Fibonacci numbers with Binet's formula?
8. How does Binet's formula compare to other methods for calculating Fibonacci numbers?

**Tip**: Binet's formula provides an exact solution for Fibonacci numbers, but for large values of \( n \), numerical precision can be an issue.

Learn how to calculate the 13th Fibonacci number using Binet's formula step-by-step. Binet's formula involves the golden ratio and provides an exact solution for Fibonacci numbers. This problem is suitable for advanced high school students interested in number theory and mathematical formulas.

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