Exam 323 - Mathematics: Problem Solving and Sequences, Exams of Mathematics

Download Exam 323 - Mathematics: Problem Solving and Sequences and more Exams Mathematics in PDF only on Docsity! Math 323 Exam # 2 – Friday July 30, 2004 Partial credit is possible, but you must show all work. Name: I hereby testify that this is individual work. Signed: 1. (a) For what values of n in the field Z5 is n3 + n2 + n + 1 = 0? (b) Solve in the field Z7 the cubic equation n3 − 3n2 − 3 = 0 2 2. Let a 6= 1 and b, be given fixed real numbers. Define the sequence (xn)n∈N by: x0 = 0 and xn+1 = axn + b (a) Define yn = xn + b a− 1 . Show that (yn)n∈N is a geometric sequence. (b) Find explicit formulas for yn and xn in terms of n. (c) For what values of a is the sequence (xn)n∈N convergent? 5 5. Let (xn)n∈N be the sequence defined by: x1 = −1 and xn+1 = 4xn + 4 xn + 6 . (a) Show that for all n ∈ N, −3 < xn < 1. (b) Show that for all n ∈ N, xn+1 − xn > 0. (c) Is the sequence (xn)n∈N convergent? If so, what is its limit? 6 5 bis. Alternates to 5 (b) (c) (d) Find α and β so that the sequence defined by yn = xn + α xn + β . is a geometric sequence. Find an explicit formula for yn in terms of n. (e) Find an explicit expression xn in terms of n, and find its limit (if it exists).