Using Binets, formula, solve for F50. (F sub 50)

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.

Learn how to calculate the 50th Fibonacci number (F50) using Binet's formula, which involves the golden ratio and properties of the Fibonacci sequence. This problem explores advanced mathematical concepts suitable for high school students in Grades 10-12.

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