Math Problem Statement

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

Solution

To solve the Fibonacci sequence for n=17n = 17 using the golden ratio formula, we use the following formula, which approximates the nn-th Fibonacci number FnF_n:

Fn=ϕn(1ϕ)n5F_n = \frac{\phi^n - (1-\phi)^n}{\sqrt{5}}

Where:

  • ϕ\phi (phi) is the golden ratio, approximately 1.6181.618,
  • 1ϕ1-\phi is approximately 0.618-0.618,
  • 5\sqrt{5} is approximately 2.2362.236.

Given n=17n = 17 and ϕ1.618\phi \approx 1.618:

  1. Calculate ϕ17\phi^{17}.
  2. Calculate (1ϕ)17(1 - \phi)^{17}.
  3. Subtract (1ϕ)17(1 - \phi)^{17} from ϕ17\phi^{17}.
  4. Divide the result by 5\sqrt{5}.

Let's calculate this step by step.

F17=1.61817(0.618)175F_{17} = \frac{1.618^{17} - (-0.618)^{17}}{\sqrt{5}}

Let's compute this to find F17F_{17}.Using the golden ratio formula, the 17th Fibonacci number is approximately F17=1596F_{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 nn?
  4. Can you derive the golden ratio formula for the Fibonacci sequence?
  5. How does the approximation improve as nn increases?

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

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Math Problem Analysis

Mathematical Concepts

Number Theory
Fibonacci Sequence
Golden Ratio

Formulas

F_n = (phi^n - (1 - phi)^n) / sqrt(5)

Theorems

Binet's Fibonacci Number Formula

Suitable Grade Level

Grades 11-12