Math Problem Statement

Using Binet's formula, find f30+f35

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:

Fn=15(αnβn)F_n = \frac{1}{\sqrt{5}} \left( \alpha^n - \beta^n \right) where:

  • α=1+52\alpha = \frac{1 + \sqrt{5}}{2} (the golden ratio)
  • β=152\beta = \frac{1 - \sqrt{5}}{2}

Now, to find F30F_{30} and F35F_{35}, we'll apply Binet's formula for n=30n = 30 and n=35n = 35 respectively. Then, we can calculate F30+F35F_{30} + F_{35}.

Let's go ahead and calculate this.Using Binet's formula:

  • F30832,040F_{30} \approx 832,040
  • F359,227,465F_{35} \approx 9,227,465

Thus, F30+F3510,059,505F_{30} + F_{35} \approx 10,059,505.

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:

  1. What is the derivation of Binet's formula?
  2. How does Binet's formula approximate Fibonacci numbers for large values?
  3. Can Binet's formula be used for non-integer indices?
  4. What is the significance of the golden ratio in Fibonacci sequences?
  5. 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)