Fibonacci Sequence Exploration

Fibonacci Sequence Exploration

Read each question carefully and provide clear, concise answers. Show all your work for full credit.

1. The Fibonacci sequence is defined by the recurrence relation F(n) = F(n-1) + F(n-2), with initial conditions F(0) = 0 and F(1) =  .

2. The ratio of consecutive Fibonacci numbers approaches the  , often denoted by the Greek letter phi (φ).

3. Fibonacci numbers appear in various natural phenomena, such as the branching of trees and the arrangement of   on a sunflower.

1. List the first 10 terms of the Fibonacci sequence, starting with F(0) = 0 and F(1) = 1.

2. Explain in your own words what a recursive definition means in the context of sequences.

1. Which of the following is the 7th term of the Fibonacci sequence (F(0)=0, F(1)=1)?

2. The Golden Ratio (φ) is approximately equal to:

1. The Fibonacci sequence can only be generated using its recursive definition.

2. The Golden Ratio is an irrational number.

1. The explicit formula for the nth Fibonacci number is given by Binet's Formula: F(n) = (φ^n - (1-φ)^n) / √5, where φ is the Golden Ratio (approximately 1.618). Use this formula to find the 5th Fibonacci number, rounding to the nearest whole number. Show your calculations.

2. Research and describe one real-world application of the Fibonacci sequence or the Golden Ratio in art, architecture, or biology. Be specific and provide an example.