Mathematical Induction

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Mathematical Induction
An introduction to proofs

NC Standard Course of Study
Competency Goal 3: The learner will describe and use recursively-defined relationships to solve problems. Objective 3.01 Use recursion to model and solve problems. Find the sum of a finite sequence. Find the sum of an infinite sequence. Determine whether a given series converges or diverges. Write explicit definitions using iterative processes, including finite differences and arithmetic and geometric formulas. Verify an explicit definition with inductive proof.

How To Verify Patterns This lesson is concerned with the way that certain kinds of patterns are verified. Because the prediction made by patterns can be erroneous and can result in the expenditure of unnecessary effort and money, it is necessary that they be as accurate as possible. The reasoning method used to verify some patterns is mathematical induction.

Mathematical Induction
This method is used to prove that certain types of discrete patterns continue. For example, with the cake division by the cut-and-choose method, the method can continue indefinitely. Initially, it began by looking at a situation with two people and then the method was extended to 3, 4 and more people. In each example, the method requires that all but one person cut and the last person choose.

Extending the Method When considering four people, the method requires that three of the people divide their piece into four pieces, and that the fourth person choose the three pieces they want. The cutters must feel that they are left with three portions that are each one-fourth of their original share, which is at least one-third of the cake.

The Fourth Person Although the fourth person may not feel that each of the three portions is at least one-third of the cake, s/he must feel that the total value of the three portions is 1. Suppose the fourth person assigns values p1, p2, and p3 to the three portions. Then p1 + p2 + p3 =1.

The Fourth Person (cont’d)
Because the fourth person is given the first choice of a portion from each of the original three people, s/he will place a value of at least on the resulting portion. Accordingly, or one-fourth of the entire cake.

Extending the Method Does this method work with 5 people, 6 people, 7 people, 8 people, ………? Yes, it does! The fact that it works is based on the mathematical principle of induction: Mathematical induction generalizes this pattern of solutions by proving that it is always possible to extend the solution to a group that is one larger than the previous. The generalization is achieved by using a variable rather than a specific number.

Dividing the Cake Suppose you know how to divide a cake fairly among k people. You need to show that it is also possible to divide a cake fairly among k + 1 people. This shows that the two-person solution can be extended to more than two people.

The proof By applying the assumption that k people can fairly divide the cake, then each person must divide their cake into k + 1 portions, that each feels are equal. The k + 1st person then selects one portion from each. Then it must be proved that this results in a share of at least 1/ (k +1) for each of the k +1 people.

The Proof (cont’d) Of the k k people who cut the cake, each should feel that each portion is 1/(k+1) of at least 1/k of the cake. Multiplying those gives 1/ (( k + 1 )k). Each person gets to keep k of the k + 1 portions, which gives a total value of at least k (1/((k+1)k) = 1/ (k+1).

The Proof (cont’d) Although the chooser may not feel that all of the original k portions are at least 1/k of the cake, s/he must feel that the total value is 1. If the person assigns values of p1, p2, …..pk to the k pieces, then p1 + p2 + … +pk = 1.

The Proof (cont’d) Because the chooser chooses first s/he is willing to place a value of at least on the resulting portion. By factoring out the 1/(k+1) and since p1 + p2 ……+ pk = 1 then each person gets 1/(k+1) of the cake.

Using Mathematical Induction
The proof is complete since it shows that whenever a cake is divided fairly among k people, it can also be divided fairly among k + 1 people. Mathematical induction is frequently used to verify that an observed formula always works.

An Example of Induction
Luis and Britt are investigating the number of handshakes that will be made by a group of people if each person shakes hands with every other person. Luis notes that if there is only one person, no handshakes are possible and that if there are two people, only one handshake is possible.

Number of People in Group
Example (cont’d) This information can be represented either by a graph or a table as shown below: 3 Number of People in Group Number of Handshakes 1 2 3 1 2

Practice Problems To use mathematical induction, you must be able to use symbols to express numeric patterns. Some of the expressions you write in this exercise will be used in the mathematical induction proof. a. If there are three people in a group and another person joins the group, there will be four people in the group. If a person leaves the original group of three, there will be two. Write expressions for the number of people if there are k people in a group and another person joins. Do the same if a person leaves the group of k people.

Practice Problems (cont’d)
Repeat this exercise for a group of k + 1 people, and then for a group of 2k people. Draw a graph like Britt’s and a table like Luis’s. a. Add another vertex to the graph to represent a fourth person, and draw segments to represent the additional handshakes that will result if the group grows to four people. Determine the number of handshakes in a group of four by adding the number of new handshakes to the number for a group of three given in the table. Write in your table the total number of handshakes for a group of four people.

Practice Problems (cont’d)
b. Add a fifth vertex to represent a fifth person, and draw segments to represent the additional handshakes. Add the number of new handshakes to the number for a group of 4 given in the table. Write in your table the total number of handshakes for a group of 5 people.

Practice Problems (cont’d)
a. Suppose that there are seven people in a group and each of them has shaken hands with every other person. If an eighth person enters the group, how many additional handshakes must be made? b. Suppose that there are k people in a group and each of them has shaken hands with every other person. If a new person enters the group, how many additional handshakes must be made?

Practice Problems (cont’d)
After studying the data for a while, Britt wonders whether the number of handshakes in a group can be found by multiplying the number of people in the group by the number that is 1 less than that and dividing this product by 2. a. If her guess is correct, how many handshakes would there be in a group of 10 people?

Practice Problems (cont’d)
Write an expression for the number of handshakes based on Britt’s guess if there are k people in a group. Do the same for a group of k + 1 people.

Recurrence Relations Britt’s formula, if correct, is sometimes known as a solution of the recurrence relation. A recurrence relation is a verbal or symbolic statement that describes how one number in a list can be derived from the previous number.

Recurrence Relations (cont’d)
One of the advantages of a recurrence relation is that it allows you to determine the number of handshakes in a group without using the number of handshakes in a smaller group. Let Hn represent the number of handshakes in a group of n people, what is the recurrence relation that expresses the relationship between Hn and Hn-1? Write the recurrence relation that expresses the relationship between Hn+1 and Hn.

Checking Britt’s Guess
To prove that Britt’s guess is correct, show that whenever the solution is known to work, it is possible to extend it to a group that is 1 larger. In other word, whenever the conjecture works for a group of k people, it will also work for a group of k + 1 people.

Practice Problems (cont’d)
Assume that Britt’s formula works for a group of k people, and write the formula for such a group. You need to show that Britt’s formula works for a group of k + 1 people. Write the formula for k + 1 people. If an additional person enters a group of k people, how many new handshakes are necessary?

Total Number of Handshakes
An expression for the total number of handshakes in a group of k + 1 people can be found by adding the expression for the number of handshakes in a group of k people (part c) to the number of new handshakes (part e):

Proof You can conclude that Britt’s formula will always work if this expression matches the one in part d. Use algebra to transform the expression until it matches the one you wrote in part d.

Practice Problems (cont’d)
Although Britt’s formula is for the number of handshakes in a group of people, it could also represent the number of potential two-party conflicts in a group. a. Use the formula to compare the number of potential conflicts when the size of a group doubles. Does the number of potential disputes also double?

Practice Problems (cont’d)
b. Why do the results of Exercise 4 suggest that some of the costs associated with government, such as that of maintaining a police force, may outpace the growth of a population?

Beginning a Proof In Exercises 1-4 you supplied several of the steps of the mathematical induction proof that began in the lesson. In Exercise 6, you will again supply many of the steps of the induction process, which requires a number of preliminary steps leading to the guessing of a formula, which must be proved.

Preliminary Steps The preliminary steps are summarized here:
Organize a table of data for several small values. For example, how many ways of voting are there with 1, 2, 3, or 4 choices on the ballot? Investigate the problem and the data to describe the pattern of the data with a recurrence relation. For example, how many ways of voting are added when another choice is placed on the ballot?

Prelim. Steps (cont’d) Make up a formula that predicts the outcome for a collection of k items. For example, what is a formula that predicts the number of ways of voting when there are k choices on the ballot? Verify that your formula works for the small values you have tabulated.

Practice Problems (cont’d)
6. Let’s look at an approval voting situation. Let’s use mathematical induction to verify that a suspected formula for the number of ways of voting under the approval system when there are n choices on the ballot is indeed correct. Number of Choices on the Ballot 1 2 3 4 Number of Possible Ways of Voting 8 16