Homework Exercise 1

Due: by 10am on Thu 17 Jan
Worth: 1.5%

For each question, please write your answer formally following the detailed format given in lecture (in particular, define P(n) carefully). Keep in mind that approximately half of the marks for each question will be given purely for having the correct proof structure, written up clearly and precisely, irrespective of the correctness of the proof.

  1. What is the value of ∑i=0,1,...,n 2i+1 for each n ∈ N? Make a conjecture and prove it formally.

  2. What amounts of postage is it possible to make using only 3¢ and 8¢ stamps? Make a conjecture and prove it formally.
    (Note: Give only one proof, using any one of the methods outlined in the lecture notes — the goal here is to make sure you understand how to prove this type of question, not to have you blindly "plug in" numbers into some other proof with no understanding of its structure or how it works...)