Mathematical Induction

Mathematical Mathematical InductionInduction

An introduction to proofsAn introduction to proofs

NC Standard Course of NC Standard Course of StudyStudy Competency Goal 3Competency Goal 3: The learner will : The learner will

describe and use recursively-defined describe and use recursively-defined relationships to solve problems. relationships to solve problems.

Objective 3.01Objective 3.01 Use recursion to model and Use recursion to model and solve problems.solve problems.a.a. Find the sum of a finite sequence. Find the sum of a finite sequence. b.b. Find the sum of an infinite sequence. Find the sum of an infinite sequence. c.c. Determine whether a given series converges or Determine whether a given series converges or

diverges. diverges. d.d. Write explicit definitions using iterative processes, Write explicit definitions using iterative processes,

including finite differences and arithmetic and including finite differences and arithmetic and geometric formulas. geometric formulas.

e.e. Verify an explicit definition with inductive proof.Verify an explicit definition with inductive proof.

How To Verify PatternsHow To Verify Patterns

This lesson is concerned with the way This lesson is concerned with the way that certain kinds of patterns are verified.that certain kinds of patterns are verified.

Because the prediction made by patterns Because the prediction made by patterns can be erroneous and can result in the can be erroneous and can result in the expenditure of unnecessary effort and expenditure of unnecessary effort and money, it is necessary that they be as money, it is necessary that they be as accurate as possible.accurate as possible.

The reasoning method used to verify The reasoning method used to verify some patterns is some patterns is mathematical mathematical induction.induction.

Mathematical Mathematical InductionInduction This method is used to prove that certain This method is used to prove that certain

types of discrete patterns continue.types of discrete patterns continue. For example, with the cake division by For example, with the cake division by

the cut-and-choose method, the method the cut-and-choose method, the method can continue indefinitely.can continue indefinitely.

Initially, it began by looking at a Initially, it began by looking at a situation with two people and then the situation with two people and then the method was extended to 3, 4 and more method was extended to 3, 4 and more people.people.

In each example, the method requires In each example, the method requires that all but one person cut and the last that all but one person cut and the last person choose.person choose.

Extending the MethodExtending the Method

When considering four people, the When considering four people, the method requires that three of the method requires that three of the people divide their piece into four people divide their piece into four pieces, and that the fourth person pieces, and that the fourth person choose the three pieces they want. choose the three pieces they want.

The cutters must feel that they are The cutters must feel that they are left with three portions that are each left with three portions that are each one-fourth of their original share, one-fourth of their original share, which is at least one-third of the which is at least one-third of the cake. cake. 3

4

1

3

1

4

The Fourth PersonThe Fourth Person

Although the fourth person may not Although the fourth person may not feel that each of the three portions is feel that each of the three portions is at least one-third of the cake, s/he at least one-third of the cake, s/he must feel that the total value of the must feel that the total value of the three portions is 1.three portions is 1.

Suppose the fourth person assigns Suppose the fourth person assigns values pvalues p11, p, p22, and p, and p33 to the three to the three portions.portions.

Then pThen p11 + p + p22 + p + p33 =1. =1.

The Fourth Person The Fourth Person (cont’d)(cont’d) Because the fourth person is given Because the fourth person is given

the first choice of a portion from each the first choice of a portion from each of the original three people, s/he will of the original three people, s/he will place a value of at least place a value of at least

on the resulting portion.on the resulting portion.

Accordingly, Accordingly, or one-fourth of the entire cake. or one-fourth of the entire cake.

1

4

1

4

1

41 2 3p p p

1

4

1

4

1

41 2 3p p p 1

4

1

411 2p p p b g ( )

Extending the MethodExtending the Method

Does this method work with 5 people, 6 Does this method work with 5 people, 6 people, 7 people, 8 people, ………?people, 7 people, 8 people, ………?

Yes, it does!Yes, it does! The fact that it works is based on the The fact that it works is based on the

mathematical principle of induction:mathematical principle of induction:Mathematical induction generalizes this Mathematical induction generalizes this pattern of solutions by proving that it is pattern of solutions by proving that it is always possible to extend the solution to always possible to extend the solution to a group that is one larger than the a group that is one larger than the previous. The generalization is achieved previous. The generalization is achieved by using a variable rather than a specific by using a variable rather than a specific number.number.

Dividing the CakeDividing the Cake

Suppose you know how to divide Suppose you know how to divide a cake fairly among k people. a cake fairly among k people. You need to show that it is also You need to show that it is also possible to divide a cake fairly possible to divide a cake fairly among k + 1 people. among k + 1 people.

This shows that the two-person This shows that the two-person solution can be extended to more solution can be extended to more than two people.than two people.

The proofThe proof

By applying the assumption that k By applying the assumption that k people can fairly divide the cake, then people can fairly divide the cake, then each person must divide their cake into each person must divide their cake into k + 1 portions, that each feels are equal.k + 1 portions, that each feels are equal.

The k + 1The k + 1stst person then selects one person then selects one portion from each.portion from each.

Then it must be proved that this results Then it must be proved that this results in a share of at least 1/ (k +1) for each of in a share of at least 1/ (k +1) for each of the k +1 people.the k +1 people.

The Proof (cont’d)The Proof (cont’d)

Of the k k people who cut the cake, Of the k k people who cut the cake, each should feel that each portion is each should feel that each portion is 1/(k+1) of at least 1/k of the cake.1/(k+1) of at least 1/k of the cake.

Multiplying those gives 1/ (( k + 1 )k).Multiplying those gives 1/ (( k + 1 )k). Each person gets to keep k of the k + Each person gets to keep k of the k +

1 portions, which gives a total value of 1 portions, which gives a total value of at least k (1/((k+1)k) = 1/ (k+1).at least k (1/((k+1)k) = 1/ (k+1).


Although the chooser may not feel Although the chooser may not feel that all of the original k portions that all of the original k portions are at least 1/k of the cake, s/he are at least 1/k of the cake, s/he must feel that the total value is 1.must feel that the total value is 1.

If the person assigns values of pIf the person assigns values of p11, , pp22, …..p, …..pkk to the k pieces, then p to the k pieces, then p11 + + pp22 + … +p + … +pkk = 1. = 1.


Because the chooser chooses first s/he Because the chooser chooses first s/he is willing to place a value of at least is willing to place a value of at least

on the resulting portion.on the resulting portion. By factoring out the 1/(k+1) and since By factoring out the 1/(k+1) and since

pp11 + p + p22 ……+ p ……+ pkk = 1 then each person = 1 then each person gets 1/(k+1) of the cake.gets 1/(k+1) of the cake.

1

1

1

1

1

11 2kp

kp

kpk

.........

Using Mathematical Using Mathematical InductionInduction The proof is complete since it The proof is complete since it

shows that whenever a cake is shows that whenever a cake is divided fairly among k people, it divided fairly among k people, it can also be divided fairly among k can also be divided fairly among k + 1 people.+ 1 people.

Mathematical induction is Mathematical induction is frequently used to verify that an frequently used to verify that an observed formula always works.observed formula always works.

An Example of An Example of InductionInduction Luis and Britt are investigating Luis and Britt are investigating

the number of handshakes that the number of handshakes that will be made by a group of people will be made by a group of people if each person shakes hands with if each person shakes hands with every other person.every other person.

Luis notes that if there is only one Luis notes that if there is only one person, no handshakes are person, no handshakes are possible and that if there are two possible and that if there are two people, only one handshake is people, only one handshake is possible.possible.

Example (cont’d)Example (cont’d)

This information can be This information can be represented either by a graph or represented either by a graph or a table as shown below:a table as shown below:

Number of Number of People in People in

GroupGroup

Number of Number of HandshakeHandshake

ss

11 00

22 11

33 33

1 2

3

Practice ProblemsPractice Problems

1.1. To use mathematical induction, you To use mathematical induction, you must be able to use symbols to must be able to use symbols to express numeric patterns. Some of express numeric patterns. Some of the expressions you write in this the expressions you write in this exercise will be used in the exercise will be used in the mathematical induction proof.mathematical induction proof.

a.a. If there are three people in a group and If there are three people in a group and another person joins the group, there will another person joins the group, there will be four people in the group. If a person be four people in the group. If a person leaves the original group of three, there leaves the original group of three, there will be two. Write expressions for the will be two. Write expressions for the number of people if there are k people in number of people if there are k people in a group and another person joins. Do the a group and another person joins. Do the same if a person leaves the group of k same if a person leaves the group of k people.people.

Practice Problems Practice Problems (cont’d)(cont’d)b.b. Repeat this exercise for a group of k + 1 Repeat this exercise for a group of k + 1

people, and then for a group of 2k people.people, and then for a group of 2k people.2.2. Draw a graph like Britt’s and a table like Draw a graph like Britt’s and a table like

Luis’s.Luis’s.a. Add another vertex to the graph to a. Add another vertex to the graph to represent a fourth person, and draw represent a fourth person, and draw segments to represent the additional segments to represent the additional handshakes that will result if the group handshakes that will result if the group grows to four people. Determine the grows to four people. Determine the number of handshakes in a group of four by number of handshakes in a group of four by adding the number of new handshakes to adding the number of new handshakes to the number for a group of three given in the the number for a group of three given in the table. Write in your table the total number of table. Write in your table the total number of handshakes for a group of four people. handshakes for a group of four people.

Practice Problems Practice Problems (cont’d)(cont’d)

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


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


4.4. After studying the data for a while, After studying the data for a while, Britt wonders whether the number of Britt wonders whether the number of handshakes in a group can be found by handshakes in a group can be found by multiplying the number of people in multiplying the number of people in the group by the number that is 1 less the group by the number that is 1 less than that and dividing this product by than that and dividing this product by 2.2.

a.a. If her guess is correct, how many If her guess is correct, how many handshakes would there be in a group handshakes would there be in a group of 10 people?of 10 people?


b.b. Write an expression for the Write an expression for the number of handshakes based on number of handshakes based on Britt’s guess if there are k Britt’s guess if there are k people in a group. Do the same people in a group. Do the same for a group of k + 1 people.for a group of k + 1 people.

Recurrence RelationsRecurrence Relations

Britt’s formula, if correct, is Britt’s formula, if correct, is sometimes known as a sometimes known as a solution solution of the recurrence relationof the recurrence relation. .

A recurrence relation is a verbal A recurrence relation is a verbal or symbolic statement that or symbolic statement that describes how one number in a describes how one number in a list can be derived from the list can be derived from the previous number.previous number.

Recurrence Relations Recurrence Relations (cont’d)(cont’d)

One of the advantages of a recurrence One of the advantages of a recurrence relation is that it allows you to determine relation is that it allows you to determine the number of handshakes in a group the number of handshakes in a group without using the number of handshakes without using the number of handshakes in a smaller group. in a smaller group.

Let HLet Hnn represent the number of represent the number of handshakes in a group of n people, what handshakes in a group of n people, what is the recurrence relation that expresses is the recurrence relation that expresses the relationship between Hthe relationship between Hnn and H and Hn-1n-1? ? Write the recurrence relation that Write the recurrence relation that expresses the relationship between Hexpresses the relationship between Hn+1n+1 and Hand Hnn..

Checking Britt’s GuessChecking Britt’s Guess

To prove that Britt’s guess is correct, To prove that Britt’s guess is correct, show that whenever the solution is show that whenever the solution is known to work, it is possible to known to work, it is possible to extend it to a group that is 1 larger. extend it to a group that is 1 larger.

In other word, whenever the In other word, whenever the conjecture works for a group of k conjecture works for a group of k people, it will also work for a group people, it will also work for a group of k + 1 people.of k + 1 people.


c.c. Assume that Britt’s formula works for Assume that Britt’s formula works for a group of k people, and write the a group of k people, and write the formula for such a group.formula for such a group.

d.d. You need to show that Britt’s formula You need to show that Britt’s formula works for a group of k + 1 people. works for a group of k + 1 people. Write the formula for k + 1 people.Write the formula for k + 1 people.

e.e. If an additional person enters a group If an additional person enters a group of k people, how many new of k people, how many new handshakes are necessary?handshakes are necessary?

Total Number of Total Number of HandshakesHandshakes An expression for the total number of An expression for the total number of

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

k kk

( )

1

2

ProofProof

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


5.5. Although Britt’s formula is for the Although Britt’s formula is for the number of handshakes in a group of number of handshakes in a group of people, it could also represent the people, it could also represent the number of potential two-party conflicts number of potential two-party conflicts in a group.in a group.

a.a. Use the formula to compare the Use the formula to compare the number of potential conflicts when the number of potential conflicts when the size of a group doubles. Does the size of a group doubles. Does the number of potential disputes also number of potential disputes also double? double?


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

Beginning a ProofBeginning a Proof

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

Preliminary StepsPreliminary Steps

The preliminary steps are summarized The preliminary steps are summarized here:here:

1.1. Organize a table of data for several Organize a table of data for several small values. For example, how many small values. For example, how many ways of voting are there with 1, 2, 3, ways of voting are there with 1, 2, 3, or 4 choices on the ballot?or 4 choices on the ballot?

2.2. Investigate the problem and the data Investigate the problem and the data to describe the pattern of the data to describe the pattern of the data with a recurrence relation. For with a recurrence relation. For example, how many ways of voting example, how many ways of voting are added when another choice is are added when another choice is placed on the ballot?placed on the ballot?

Prelim. Steps (cont’d)Prelim. Steps (cont’d)

3.3. Make up a formula that predicts the Make up a formula that predicts the outcome for a collection of k items. outcome for a collection of k items. For example, what is a formula that For example, what is a formula that predicts the number of ways of predicts the number of ways of voting when there are k choices on voting when there are k choices on the ballot?the ballot?

4.4. Verify that your formula works for Verify that your formula works for the small values you have the small values you have tabulated.tabulated.


6.6. Let’s look at an approval voting Let’s look at an approval voting situation. Let’s use mathematical situation. Let’s use mathematical induction to verify that a suspected induction to verify that a suspected formula for the number of ways of formula for the number of ways of voting under the approval system voting under the approval system when there are n choices on the when there are n choices on the ballot is indeed correct.ballot is indeed correct.Number of Choices on the BallotNumber of Choices on the Ballot 11 22 33 44

Number of Possible Ways of Number of Possible Ways of VotingVoting 22 44 88 1616

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Mathematical Induction