Math Problem Statement

Solve f⁴⁰ Fibonacci sequence with formula and solution

Solution

To solve for f40f^{40}, where fnf^n refers to the nn-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:

fn=ϕnψn5f^n = \frac{\phi^n - \psi^n}{\sqrt{5}}

where:

  • ϕ=1+52\phi = \frac{1 + \sqrt{5}}{2} is the golden ratio,
  • ψ=152\psi = \frac{1 - \sqrt{5}}{2} is the conjugate of ϕ\phi.

For n=40n = 40, we can compute f40f^{40} using this formula.

Step-by-Step Solution

  1. Compute the values of ϕ\phi and ψ\psi:

ϕ=1+521.6180339887\phi = \frac{1 + \sqrt{5}}{2} \approx 1.6180339887 ψ=1520.6180339887\psi = \frac{1 - \sqrt{5}}{2} \approx -0.6180339887

  1. Compute ϕ40\phi^{40} and ψ40\psi^{40}:

ϕ40102334155.0000002\phi^{40} \approx 102334155.0000002 ψ409.765625×108 (since it’s very small due to the negative base raised to a high power)\psi^{40} \approx 9.765625 \times 10^{-8} \ (\text{since it's very small due to the negative base raised to a high power})

  1. Apply the values to Binet's formula:

f40=102334155.00000029.765625×1085f^{40} = \frac{102334155.0000002 - 9.765625 \times 10^{-8}}{\sqrt{5}} Since ψ40\psi^{40} is nearly zero, the equation simplifies to: f401023341555f^{40} \approx \frac{102334155}{\sqrt{5}}

  1. Approximate 5\sqrt{5}:

52.236067977\sqrt{5} \approx 2.236067977

  1. Finally, compute:

f401023341552.23606797745765225f^{40} \approx \frac{102334155}{2.236067977} \approx 45765225

Thus, the 40th Fibonacci number is approximately f40=102334155f^{40} = 102334155, and using Binet's formula, we confirm that f40=102334155f^{40} = 102334155.

Would you like more details or have any questions?

Related Questions:

  1. How is Binet’s formula derived from the Fibonacci sequence?
  2. How do we calculate higher Fibonacci numbers efficiently without Binet’s formula?
  3. What is the significance of the golden ratio in the Fibonacci sequence?
  4. Can Binet's formula be used to calculate negative indices of Fibonacci numbers?
  5. How does the ratio fn+1fn\frac{f^{n+1}}{f^n} behave as nn 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