Math Problem Statement
Solve f⁴⁰ Fibonacci sequence with formula and solution
Solution
To solve for , where refers to the -th Fibonacci number, we can use the Binet’s Formula for Fibonacci numbers. This formula expresses the Fibonacci numbers in terms of powers of irrational numbers:
where:
- is the golden ratio,
- is the conjugate of .
For , we can compute using this formula.
Step-by-Step Solution
- Compute the values of and :
- Compute and :
- Apply the values to Binet's formula:
Since is nearly zero, the equation simplifies to:
- Approximate :
- Finally, compute:
Thus, the 40th Fibonacci number is approximately , and using Binet's formula, we confirm that .
Would you like more details or have any questions?
Related Questions:
- How is Binet’s formula derived from the Fibonacci sequence?
- How do we calculate higher Fibonacci numbers efficiently without Binet’s formula?
- What is the significance of the golden ratio in the Fibonacci sequence?
- Can Binet's formula be used to calculate negative indices of Fibonacci numbers?
- How does the ratio behave as approaches infinity?
Tip:
For large Fibonacci numbers, Binet’s formula provides an efficient way to approximate values directly, without the need for recursive computation.
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Math Problem Analysis
Mathematical Concepts
Fibonacci Sequence
Golden Ratio
Algebra
Formulas
Binet’s Formula: f^n = (φ^n - ψ^n) / √5
Golden Ratio (φ) = (1 + √5) / 2
Conjugate (ψ) = (1 - √5) / 2
Theorems
Binet’s Theorem
Properties of the Fibonacci Sequence
Suitable Grade Level
Grades 9-12