Math Problem Statement
Using Binets’ formula, solve for F50. (F sub 50
Solution
Binet's formula is used to find the -th Fibonacci number without the need to compute all the preceding numbers. The Fibonacci sequence is defined as:
where and .
Binet's formula is expressed as:
where:
- (the golden ratio)
- (its conjugate)
Now, to calculate , plug into the formula:
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 using Binet's formula with a calculator or any software that can handle large exponents.
- Compute , where .
- Compute , where .
- Subtract from .
- Divide the result by , which is approximately 2.23607.
Do you want further details on any step? Here are some related questions:
- How can I calculate powers of irrational numbers manually?
- What is the significance of the golden ratio () in Fibonacci numbers?
- How does Binet's formula approximate large Fibonacci numbers?
- Can Binet’s formula be used for small Fibonacci numbers too?
- How does the approximation of affect the accuracy of large Fibonacci numbers?
Tip: When using Binet's formula for large Fibonacci numbers, the contribution from becomes negligible, so alone gives a good approximation.
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Math Problem Analysis
Mathematical Concepts
Fibonacci Sequence
Exponential Functions
Golden Ratio
Formulas
Binet's Formula: F_n = (phi^n - psi^n) / sqrt(5)
Golden Ratio: phi = (1 + sqrt(5)) / 2
Psi: psi = (1 - sqrt(5)) / 2
Theorems
Binet's Formula
Suitable Grade Level
Grades 10-12