Fibonacci Identities

MATH 242 practice problems

Learning Goals: Formulate precise mathematical conjectures supported by computational evidence.

Your task: Complete Exercises 2.21, 2.22, and 2.23 from our Experimental Mathematics text. These exercises are reprinted below.

Due: Friday, February 27, at 5pm. Upload your solution as a Mathematica notebook to the Fibonacci Identities assignment on Moodle.

Grading: If you state a conjecture involving Fibonacci numbers and supported by your computational evidence, you will earn a Satisfactory grade for each problem.

Need help? Review the notes from class (which are posed on the course web site), talk with your classmates, email Prof. Wright, or visit the help session.

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What assistance did you receive and what resources did you use (if any) while working on this assignment?

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Problem 1: Exercise 2.21 in the text

Compute (Subscript[F, m])(Subscript[F, n+1])-(Subscript[F, m+1])(Subscript[F, n]) for lots of positive integers n and m>n. What patterns do you observe? Conjecture an identity whose left side is (Subscript[F, m])(Subscript[F, n+1])-(Subscript[F, m+1])(Subscript[F, n]).

Problem 2: Exercise 2.22 in the text

Investigate (Subscript[F, m])(Subscript[F, n+j])-(Subscript[F, m+j])(Subscript[F, n]) computationally (that is, for lots of positive integers n, m>n, and j). Based on your evidence, state a Fibonacci identity conjecture. 

(Hint: If you set j=1, then you’re back to the previous problem. So try j=2 first. Then try j=3. Then try other j.)

Problem 3: Exercise 2.23 in the text

Compute (Subscript[F, m])(Subscript[F, n])+(Subscript[F, m+1])(Subscript[F, n+1]) for lots of positive integers m and n. What patterns do you observe? Conjecture an identity whose left side is (Subscript[F, m])(Subscript[F, n])+(Subscript[F, m+1])(Subscript[F, n+1]).