Solve this Fibonacci sequence using the golden ratio formula Where; n=17 & phi=1.618

To solve the Fibonacci sequence for \( n = 17 \) using the golden ratio formula, we use the following formula, which approximates the \( n \)-th Fibonacci number \( F_n \):

\[
F_n = \frac{\phi^n - (1-\phi)^n}{\sqrt{5}}
\]

Where:
- \( \phi \) (phi) is the golden ratio, approximately \( 1.618 \),
- \( 1-\phi \) is approximately \( -0.618 \),
- \( \sqrt{5} \) is approximately \( 2.236 \).

Given \( n = 17 \) and \( \phi \approx 1.618 \):

1. Calculate \( \phi^{17} \).
2. Calculate \( (1 - \phi)^{17} \).
3. Subtract \( (1 - \phi)^{17} \) from \( \phi^{17} \).
4. Divide the result by \( \sqrt{5} \).

Let's calculate this step by step.

\[
F_{17} = \frac{1.618^{17} - (-0.618)^{17}}{\sqrt{5}}
\]

Let's compute this to find \( F_{17} \).Using the golden ratio formula, the 17th Fibonacci number is approximately \( F_{17} = 1596 \).

Would you like more details or have any questions about this process?

Here are 5 related questions to expand on this topic:
1. How does the golden ratio relate to the Fibonacci sequence?
2. What is the significance of using the golden ratio for approximating Fibonacci numbers?
3. How accurate is the golden ratio formula for large values of \( n \)?
4. Can you derive the golden ratio formula for the Fibonacci sequence?
5. How does the approximation improve as \( n \) increases?

**Tip:** The golden ratio provides an efficient way to approximate Fibonacci numbers without computing the entire sequence, especially for larger values of \( n \).

Discover how to calculate the 17th Fibonacci number using the golden ratio formula with phi ≈ 1.618. This solution involves number theory concepts, the Fibonacci sequence, and Binet's Fibonacci formula. Ideal for high school students (Grades 11-12).

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