Fibonacci Numbers Exploration
A Grade 12 math worksheet exploring the properties and applications of Fibonacci numbers, including the Golden Ratio and recurrence relations.
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Fibonacci Numbers 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) = 1. The first five terms of the sequence are 0, 1, 1, 2, 3. The next three terms are , , and .
2. The ratio of consecutive Fibonacci numbers approaches a special irrational number called the , denoted by the Greek letter phi (φ).
3. Which of the following statements about Fibonacci numbers is true?
Every Fibonacci number is an even number.
The sum of the first 'n' Fibonacci numbers equals F(n+2) - 1.
Fibonacci numbers only appear in mathematics and have no real-world applications.
The product of any two consecutive Fibonacci numbers is always a perfect square.
4. Calculate the 10th Fibonacci number, F(10), given F(0)=0 and F(1)=1.
5. The Golden Ratio (φ) is approximately 1.618.
True
False
6. Describe one real-world example where Fibonacci numbers or the Golden Ratio can be observed. Explain how it relates to the concept.
7. The Fibonacci spiral is constructed by drawing quarter circles in a series of squares with side lengths corresponding to Fibonacci numbers. Draw the next two squares and the corresponding quarter circles to extend the spiral shown below. Assume the smallest square has a side length of 1 unit.