8.4 Mathematical Induction 2015. Copyright © by Houghton Mifflin Company, Inc. All rights reserved....

8.4Mathematical

Induction2015

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2

Recursively defined sequences

Write the first 5 terms of the sequence beginning with the given term. Calculate the first and second differences of the sequence. Find a model for the sequence.

1 10 2n na a a n

2 2na n n


Warm-Up 10/16/2013

Find P5 for3(2 1) .1k

kP k

553[2( ) ]

151P

Replace k with 5.

Simplify.3(11)4

334 Simplify.


Warm-Up 2/14/2013

Find Pk + 1 for 3(2 1) .1kkP k

13[2( ) ]1 1

1 1kP kk

Replace k with k + 1.

Simplify.3(2 2 1)kk

3(2 3)kk Simplify.


• Does a pattern appear when adding blocks to create a square?

+…+ ( ) =

Will it always hold?

Recognizing a pattern and assuming it isalways true is not a valid method of proof.

1 3 5 7 2 1n 2n


Mathematical induction is a legitimate method of proof for all positive integers n.

Principle: Let Pn be a statement involving n, a positive integer. If

1. P1 is true, and

2. the truth of Pk implies the truth of Pk + 1 for every positive k,

then Pn must be true for all positive integers n.

Prove it by showing that: 1k kP the next term P


Example: Prove the following sum formula using mathematical induction.

• To prove a sum formula you must show that:• Sum of n terms + the next term=Sum of (n+1)terms.Ie, show that:1) It is true for n=12) assume is true3) show that

21 3 5 7 ... (2 1)nS n n

1 1n n nS a S nS


• Example (con’t):

1) 2) assume is true3) show that

21 3 5 7 ... (2 1)nS n n

1 1n n nS a S

21 1

nS

2 21(2( ) 1 1) ( )nn n

2 22 2 1 2 1n n n n 2 22 1 2 1n n n n

2 2Or 1 1n n


Example:Use mathematical induction to prove

Sn = 2 + 4 + 6 + 8 + . . . + 2n = n(n + 1)

for every positive integer n.

1. Show that the formula is true when n = 1.

S1 = n(n + 1) = 1(1 + 1) = 2 True

2. Assume the formula is valid for some integer n.

3. Use this assumption to prove the formula Sk =k(k + 1) is valid for the next integer, n + 1 and show that the formula Sk + 1 = (k + 1)(k + 2) is true.

Sk = 2 + 4 + 6 + 8 + . . . + 2k = k(k + 1) Assumption


1 1n n nS a S

Example(con’t): Use mathematical induction to prove Sn = 2 + 4 + 6 + 8 + . . . + 2n = n(n + 1) for every positive integer n.

1 2 1 1 2n n n n n

1 2n n

2 3 2n n

2 2 2n n n

The formula Sn = n(n + 1) is valid for all positive integer values of n.

Show that


You try: Prove the sum formula:

• 1) true for n=1?• 2) assume is true• 3) show that

5 7 9 11 ... (3 2 ) ( 4)nS n n n

1 1n n nS a S kS


Sums of Powers of Integers :

1

( 1) 1 2 3 41 2.n

i

n ni n

2 2 2 2 2 2

1

( 1)(2 1) 1 2 3 42. 6n

i

n n ni n

2 23 3 3 3 3 3

1

( 1) 1 2 3 43. 4n

i

n ni n

24 4 4 4 4 4

1

( 1)(2 1)(3 3 1) 1 2 3 44. 30n

i

n n n n ni n

2 2 25 5 5 5 5 5

1

( 1) (2 2 1) 1 2 3 4 15 2.n

i

n n n ni n


1

( 1)1 2 3 4 2n

i

n ni n

You Try:

Use mathematical induction to prove the following:


2 2 2 2 2 2

1

( 1)(2 1)1 2 3 4 .6n

i

n n ni n

Example:Use mathematical induction to prove for all positive integers n,

Assumption2 2 2 2 2 ( 1)(2 1)1 2 3 4 6k kS k k k

1( 1)(2( ) 1) 1(2)(2 1) 6 16

116 6

1S True

Proof:

8.4 Homework Day 1

• Pg. 592 7-21 odd, skip #15


8.4Mathematical

InductionDay 2


Precalculus HWQ 8-4 day 2

• Use mathematical induction to prove the formula:

1

12 1 2 1 2 1

n

n

nn n n


Finite DifferencesThe first differences of the sequence 1, 4, 9, 16, 25, 36 are found by subtracting consecutive terms.

n: 1 2 3 4 5 6an: 1 4 9 16 25 36

First differences: 3 5 7 9 11

Second differences: 2 2 2 2

The second differences are found by subtracting consecutive first differences.

quadratic model


When the second differences are all the same nonzero number, the sequence has a perfect quadratic model.

Find the quadratic model for the sequence1, 4, 9, 16, 25, 36, . . .

an = an2 + bn + c

a1 = a(1)2 + b(1) + c = 1a2 = a(2)2 + b(2) + c = 4

a3 = a(3)2 + b(3) + c = 9

Solving the system yields a = 1, b = 0, and c = 0.

an = n2

a + b + c = 14a + 2b + c = 4

9a + 3b + c = 9


an = an2 + bn + ca0 = a(0)2 + b(0) + c = 3a1 = a(1)2 + b(1) + c = 3a4 = a(4)2 + b(4) + c = 15

an = n2 – n + 3

c = 3a + b + c = 3

16a + 4b + c = 15

Solving the system yields a = 1, b = –1, and c = 3.

Example: Find the quadratic model for the sequence with

a0 = 3, a1 = 3, a4 = 15.


You Try:Take the differences and find a model for the sequence: 5, 8, 11, 14, …

3 2na n


• Take the differences and find a model for the sequence. Use the regression feature of your graphing calculator.

• 3, 5, 8, 12, 17, 23, …

21 1 22 2na n n


• Take the differences and find a model for the sequence. Use the regression feature of your graphing calculator.

• 2, 13, 46, 113, 226, 397, 638, 961 …

3 22 1na n n



Write the first 5 terms of the sequence beginning with the given term. Find a model for the sequence.

1 10 2n na a a n

2 2na n n

8.4 Homework Day 2

• Pg. 592 8-14 even, 37-43 odd



Precalculus Warm-Up 2/15

• Find a quadratic model for the sequence where:

1 3 78, 40, 176a a a



Write the first 5 terms of the sequence beginning with the given term. Calculate the first and second differences of the sequence. Find a model for the sequence.

1 10 2n na a a n

2 2na n n

8.4 Mathematical Induction 2015. Copyright © by Houghton Mifflin Company, Inc. All rights reserved....

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