CS 1050 B: Construction Proofs January 24, 2008
Homework 2
Lecturer: Sasha Boldyreva Due: January 31, 2008
Assignment 2.01 Do the assigned reading.
Assignment 2.02 Indicate how much time did you spend on this homework.
Problem 2.1, 5 points. Use mathematical induction to prove that 2n+ 3 ≤2nfor all
n≥4.
Problem 2.2, 5 points. Use mathematical induction to show that ndistinct lines in
the plane passing through the same point divide the plane into 2nregions.
Problem 2.3, 8 points. Let a1= 2, a2= 9, and an= 2an−1+ 3an−2for n≥3. Use
strong induction (the Second Principle of Mathematical Induction) to show that an≤3n
for all positive integers n.
Problem 2.4, 5 points. Find the error in the following proof of this “theorem”:
Theorem: Every positive integer equals the next largest positive integer.
Proof: Let P(n) be the predicate “n = n + 1”. To show that P(k)→P(k+ 1), assume
that P(k) is true for some k, so that k=k+ 1. Add 1 to both sides of this equation to
obtain k+ 1 = k+ 2, which is P(k+ 1). Therefore P(k)→P(k+ 1) is true. Hence P(n) is
true for all positive integers n.
Problem 2.5, 6 points. Sharing a chocolate bar. Problem 10 from Section 4.2 of
Rosen’s textbook.
Problem 2.6, 6 points. Describe a recursive algorithm for computing 52nwhere nis
a nonnegative integer.
Problem 2.7, 6 points. Problem 38 from Section 4.4 of Rosen’s textbook.