Lucas Identities

Math 242 practice problems

Name: _________________

Learning Goals: Develop and implement an algorithm to solve a mathematical problem, and interpret your results. Formulate precise mathematical conjectures supported by computational evidence. Explore generalizations of known mathematical ideas.

Your task: Complete Exercises 2.47, 2.53, and 2.54 from our Computational Mathematics text. These exercises are reprinted below.

Due: Monday , March 4, at 5:00pm. Upload your solution as a Mathematica notebook to the Lucas Identities assignment (Moodle link).

Grading: If you state conjectures 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.

Problem 1: Exercise 2.47

Compute (Subscript[L, n+1])(Subscript[L, n-1])-(Subscript[L, n]^2) for various positive integers n. Conjecture an identity, supported by computational evidence.

Problem 2: Exercise 2.53

Find the Lucas 3n identity of the form (Subscript[L, 3n])=a (Subscript[L, n]^3)+b (Subscript[L, n]^2)+c (Subscript[L, n])+d . First, use four even values of n to set up a linear system which you can solve for the coefficients. Then repeat for four odd value of n. Combine your expression to obtain a single formula for both even and odd n, and confirm that your identity holds for lots of values of n.

Problem 3: Exercise 2.54

Find the Lucas 4n identity of the form (Subscript[L, 4n])=(Subscript[a, 4]) (Subscript[L, n]^4)+(Subscript[a, 3]) (Subscript[L, n]^3)+(Subscript[a, 2]) (Subscript[L, n]^2)+(Subscript[a, 1])(Subscript[L, n])+(Subscript[a, 0]) . As before, consider even and odd n separately, then find a single formula that holds for both cases. Provide computational evidence that your identity holds for lots of values of n.