(F_{50}): [ F_{50} = \frac{(\phi^{50} - (1 - \phi)^{50})}{\sqrt{5}} ]

The expression you've provided represents the 50th Fibonacci number \( F_{50} \), using Binet's formula. Let's break it down:

### Binet's Formula for the Fibonacci Sequence:
\[
F_n = \frac{\phi^n - (1 - \phi)^n}{\sqrt{5}}
\]
where:
- \( \phi \) is the golden ratio, given by \( \phi = \frac{1 + \sqrt{5}}{2} \),
- \( n \) is the position of the Fibonacci number in the sequence (in your case, \( n = 50 \)).

Now, for \( F_{50} \):
\[
F_{50} = \frac{\phi^{50} - (1 - \phi)^{50}}{\sqrt{5}}
\]
This formula works due to the rapid decay of \( (1 - \phi)^n \), making it negligible for large \( n \), so the dominant term is \( \phi^n \).

Would you like me to calculate the exact value of \( F_{50} \) using this formula?

Here are five related questions to expand the topic:

1. What is the significance of the golden ratio \( \phi \) in the Fibonacci sequence?
2. Can you derive the approximation of \( F_n \) for large \( n \) based on Binet's formula?
3. How does the second term \( (1 - \phi)^n \) contribute to the accuracy of the formula?
4. How does the growth rate of Fibonacci numbers relate to the golden ratio?
5. What are some real-world applications of the Fibonacci sequence?

**Tip:** The Fibonacci sequence appears in many natural phenomena, such as the arrangement of leaves on a stem or the spirals of shells.

F_{50} = \frac{(\phi^{50} - (1 - \phi)^{50})}{\sqrt{5}}

Explore how to compute the 50th Fibonacci number using Binet's Formula. This problem covers the golden ratio, Binet's theorem, and shows how the Fibonacci sequence grows. Suitable for students in Grades 11-12 and above.

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