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 using the golden ratio formula, we use the following formula, which approximates the -th Fibonacci number :
Where:
- (phi) is the golden ratio, approximately ,
- is approximately ,
- is approximately .
Given and :
- Calculate .
- Calculate .
- Subtract from .
- Divide the result by .
Let's calculate this step by step.
Let's compute this to find .Using the golden ratio formula, the 17th Fibonacci number is approximately .
Would you like more details or have any questions about this process?
Here are 5 related questions to expand on this topic:
- How does the golden ratio relate to the Fibonacci sequence?
- What is the significance of using the golden ratio for approximating Fibonacci numbers?
- How accurate is the golden ratio formula for large values of ?
- Can you derive the golden ratio formula for the Fibonacci sequence?
- How does the approximation improve as increases?
Tip: The golden ratio provides an efficient way to approximate Fibonacci numbers without computing the entire sequence, especially for larger values of .
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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