Fibonacci Sequence Exploration
This worksheet explores the properties and applications of the Fibonacci sequence, suitable for Grade 12 students.
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Fibonacci Sequence Exploration
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Read each question carefully and provide your answers in the space provided. 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 first 8 terms of the Fibonacci sequence, starting with F(0), are 0, 1, 1, 2, 3, 5, 8, .
3. The ratio of consecutive Fibonacci numbers approaches the , often denoted by the Greek letter phi (φ).
4. Which of the following is a property of the Fibonacci sequence?
Every third Fibonacci number is even.
The sum of the first 'n' Fibonacci numbers is F(n+2) - 1.
Both a and b.
Neither a nor b.
5. What is the 10th Fibonacci number, F(9), given F(0)=0 and F(1)=1? Show your steps.
6. The Fibonacci sequence appears in nature, such as in the branching of trees and the arrangement of leaves on a stem.
True
False
7. Explain the relationship between the Fibonacci sequence and the Golden Ratio (φ).
8. Draw the first few squares of a Fibonacci spiral within the grid below. Start with two 1x1 squares.