Test Take-Home 2, Math 240, Fall 2005
Due: 2:45pm, October 4. Value: 50 pts. Instructions: In contrast to assignments, you must not discuss this take-home test with others, and you may not use electronic resources. The only written resources you may use are your textbook and class notes that you have written yourself. Be careful not to place your test where others might read your solutions. You may discuss the take-home test with me, although I wont promise to be helpful. I will schedule a time for October 5 or 6 when you will formally present at least two of your solutions to me. A portion of your grade will be based on your presentation. (If you are late for your appointment, or if you miss it, you will receive a penalty.) I recommend keeping a copy of your work so that you can study your solutions before presenting them; your presentation, though, should not involve pointing to your copy. You will have the opportunity to complete a rewrite, due the class day following when I distribute your initial grades. You will receive a three-point penalty on the rewrite, and your overall grade will be the higher of your initial grade and your rewrite grade. Your rst graded version must be submitted with your rewrite. Because of the rewrite policy, should you be careful not to discuss the test with others or access related resources even after you have submitted your initial version. Problem A. Recall that we dened fn is dened to be the nth Fibonacci number, starting with f 0 as 0 and f1 as 1. Suppose that we dene a new sequence g n dened where gn = 0 if n = 0 gn1 + fn if n > 0
Compute the rst nine values g1 , g2 , . . . g9 , and form a hypothesis relating the g n sequence to a sequence you already know. Prove (using induction) that your hypothesis is correct. Problem B. Suppose we have a bag lled with three types of Legos: 1 1 white pieces, 2 1 blue pieces, and 2 1 red pieces. Somebody has given us a n 1 green piece, and we want to place white, blue, and red pieces to cover the t...