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

Binet's formula is used to find the \(n\)-th Fibonacci number without the need to compute all the preceding numbers. The Fibonacci sequence is defined as:

\[
F_n = F_{n-1} + F_{n-2}
\]
where \(F_0 = 0\) and \(F_1 = 1\).

Binet's formula is expressed as:

\[
F_n = \frac{\phi^n - \psi^n}{\sqrt{5}}
\]
where:
- \(\phi = \frac{1 + \sqrt{5}}{2} \approx 1.61803\) (the golden ratio)
- \(\psi = \frac{1 - \sqrt{5}}{2} \approx -0.61803\) (its conjugate)

Now, to calculate \(F_{50}\), plug \(n = 50\) into the formula:

\[
F_{50} = \frac{\phi^{50} - \psi^{50}}{\sqrt{5}}
\]

Let me calculate that for you.It seems like I can’t do more advanced data analysis right now. Please try again later. However, I can guide you on how to compute \( F_{50} \) using Binet's formula with a calculator or any software that can handle large exponents.

1. Compute \( \phi^{50} \), where \( \phi = 1.61803 \).
2. Compute \( \psi^{50} \), where \( \psi = -0.61803 \).
3. Subtract \( \psi^{50} \) from \( \phi^{50} \).
4. Divide the result by \( \sqrt{5} \), which is approximately 2.23607.

Do you want further details on any step? Here are some related questions:

1. How can I calculate powers of irrational numbers manually?
2. What is the significance of the golden ratio (\(\phi\)) in Fibonacci numbers?
3. How does Binet's formula approximate large Fibonacci numbers?
4. Can Binet’s formula be used for small Fibonacci numbers too?
5. How does the approximation of \(\psi^n\) affect the accuracy of large Fibonacci numbers?

**Tip:** When using Binet's formula for large Fibonacci numbers, the contribution from \( \psi^n \) becomes negligible, so \(\phi^n\) alone gives a good approximation.

Learn how to calculate the 50th Fibonacci number using Binet's formula. This detailed explanation covers key concepts such as the Fibonacci sequence, the golden ratio, and exponential functions. Suitable for students in Grades 10-12.

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