Assignment 8 (Proofs and Mathematical Induction)

1. A Proof by Induction:
   Let P(n) be the statement that \( 1 + \frac{1}{4} + \frac{1}{9} + . . . + + \frac{1}{n^2} < 2 - \frac{1}{n} \) where n is an integer greater than 1.

   1. What is the statement P(2)?
   2. Show that P(2) is true, completing the basis step of a proof by mathematical induction that P(n) is true for all integers n greater than 1.
   3. What is the inductive hypothesis of a proof by mathematical induction that P(n) is true for all integers n greater than 1?
   4. What do you need to prove in the inductive step of a proof by mathematical induction that P(n) is true for all integers n greater than 1?
   5. Complete the inductive step of a proof by mathematical induction that P(n) is true for all integers n greater than 1.
   6. Explain why these steps show that this inequality is true whenever n is an integer greater than 1.

2. Flavors of Ice Cream: The following claims to prove that all ice cream flavors are equally tasty. Since the result is clearly false, identify the specific error in this argument. (For example, at what step does this argument fail and why.)

   Induction Hypothesis: Let P(n) be the predicate if {IC₁, IC₂, IC₃, ..., IC_n} is any set of ice cream flavors. Then each of the flavors in the set are equally tasty.
   
   For convenience in notation, we will use an equal sign (=) to indicate flavors are equally tasty.

   IC₁ = IC₂ = IC₃ = ... = IC_n

   Proof by Induction on n.

   Base case: When n = 1,

   Let {IC₁} be any set with 1 flavor of ice cream. Then, only 1 flavor is represented, so all flavors in the set is equally tasty.

   Inductive case: Assume P(n) and prove P(n+1).

   Set S = {IC1, IC2, IC3, ... ICn, ICn+1} be any set of n+1 flavors of ice cream. Now consider subsets U = {IC1, IC2, IC3, ..., ICn} V = {IC2, IC3, ..., ICn, ICn+1} Since both U and V have n flavors, we can apply the induction hypothesis P(n) to get IC1 = IC2 = IC3 = ... = ICn IC2 = IC3 = ... = ICn = ICn+1 By transitivity—combining the flavors that are equally tasty, we conclude IC1 = IC2 = IC3 = ... = ICn = ICn+1 This proves P(n+1).

3. Postage from 3-cent and 5-cent stamps:
   Prove that any amount of postage worth 8 cents or more can be made from 3-cent or 5-cent stamps.
   Note: This exercise is taken from Exercise 8.6.1a in the zyBook text.

created October 25, 2025 revised October-November, 2025
For more information, please contact Henry M. Walker at walker@cs.grinnell.edu.