Unit 1 Discussion: Fibonacci Sequence Discussion

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In Unit 1, we are learning about the Fibonacci Sequence. For this discussion, you will post at least twice – 1) an initial post, and then 2) a reply to a classmate.

The Pythagorean Theorem is used to describe the relationship among the three sides in a right triangle. The relationship is generally written as a2 + b2 = c2, where a and b are the short sides of the right triangle and c is the hypotenuse. 

                                                                                                                                      

A Pythagorean triple is a set of three whole numbers {a, b, c} that satisfy a2 + b2 = c2. For example, since 62 + 82 = (10)2, we say that {6, 8, 10} is a Pythagorean triple.

We can use the following steps to determine Pythagorean triples using any four consecutive Fibonacci terms or four consecutive Fibonacci-like terms. 

  1. Determine the product of 2 and the two inner Fibonacci numbers. This will be a.
  2. Determine the product of the two outer numbers.  This is b.
  3. Determine the sum of the squares of the inner two numbers.  This will be c.
  4. This process has produced the Pythagorean triple, {a, b, c}. You can verify by checking to see if a2 + b2 is equal to c2.

Here are two examples; Example 1 uses the Fibonacci terms 3, 5, 8 and 13. Example 2 uses the Fibonacci-like terms 1, 4, 5, 9

  1. Determine the product of 2 and the two inner Fibonacci numbers.
    • Example 1: the inner numbers are 5 and 8; the product of 2 and 5 and 8 is 80. This will be a.
    • Example 2: the inner numbers are 4 and 5; the product of 2 and 4 and 5 is 40. This will be a.
  2. Determine the product of the two outer numbers.
    • Example 1: 3 times 13 is 39. This is b.
    • Example 2: 1 times 9 is 9. This is b.
  3. Determine the sum of the squares of the inner two numbers.
    • Example 1: we have 52 + 82 = 25 + 64 = 89. This will be c.
    • Example 2: we have 42 + 52 = 16 + 25 = 41. This will be c.
  4. This process has produced the Pythagorean triples. You can verify these!
    • Example 1: {80, 39, 89}.
    • Example 2: {40, 9, 41}

For your initial post

  1. Select four (4) consecutive Fibonacci terms or four (4) consecutive Fibonacci-like terms and use those four terms to create a Pythagorean triple by following the four steps above. NOTE: You cannot use 3, 5, 8 and 13, or 1, 4, 5, 9.
  2. Verify your results using the Pythagorean theorem.

For your reply

Respond to another student’s post. Do you agree with their Pythagorean triple? How does it differ from yours?

Grading Criteria

For this discussion, you must make two postings to receive full credit:

1. Initial posting (up to 15 points): Answer the two question in the instructions. You must explain your answers by showing calculations. Each answer must include at least 1 sentence along with the calculations.

Grading criteria:

  • Correct and complete answer: 15 points
  • Incorrect and/or incomplete answer: 5 points.
  • Incorrect and incomplete: 0 points.

2. Second posting (up to 5 points): Respond to another student’s posting. Your answer must include at least 2 sentences.

Grading criteria:

  • Posting elaborates or provides additional relevant information: 5 points
  • Incomplete response: 0 points

(Learning Objectives Assessed: 1d, 1e, 1f)