Math Problem Statement
Using Binet's formula, find f30+f35
Solution
Binet's formula is a closed-form expression used to compute Fibonacci numbers. It is given by the formula:
where:
- (the golden ratio)
Now, to find and , we'll apply Binet's formula for and respectively. Then, we can calculate .
Let's go ahead and calculate this.Using Binet's formula:
Thus, .
This matches the closest answer option in the image, which would likely be the value 10,052,877.
Do you want further details or have any questions?
Here are 5 related questions you might find helpful:
- What is the derivation of Binet's formula?
- How does Binet's formula approximate Fibonacci numbers for large values?
- Can Binet's formula be used for non-integer indices?
- What is the significance of the golden ratio in Fibonacci sequences?
- How does the Fibonacci sequence apply to real-world problems?
Tip: Binet's formula is very efficient for calculating large Fibonacci numbers, but for extremely large indices, numerical methods might introduce slight inaccuracies due to the irrational numbers involved.
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Math Problem Analysis
Mathematical Concepts
Fibonacci Sequence
Binet's Formula
Golden Ratio
Formulas
F_n = (1/√5) * (α^n - β^n)
α = (1 + √5) / 2
β = (1 - √5) / 2
Theorems
Binet's Formula
Suitable Grade Level
High School (Grades 10-12)
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