1. 8.4 Mathematical Induction 2015

2. 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.

3. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 3 Example: Find P k+1 Warm-Up 10/16/2013 Find P 5 for Replace k with 5. Simplify.

4. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4 Example: Find P k+1 Warm-Up 2/14/2013 Find P k + 1 for Replace k with k + 1. Simplify.

5. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5 Does a pattern appear when adding blocks to create a square? +…+ ( ) = Will it always hold? Recognizing a pattern and assuming it is always true is not a valid method of proof.

6. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 6 Mathematical induction is a legitimate method of proof for all positive integers n. Definition: Mathematical Induction Principle: Let P n be a statement involving n, a positive integer. If 1. P 1 is true, and 2. the truth of P k implies the truth of P k + 1 for every positive k, then P n must be true for all positive integers n. Prove it by showing that:

7. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7 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=1 2) assume is true 3) show that

8. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8 Example (con’t): 1) 2) assume is true 3) show that

9. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9 Example: Using Induction to Prove a Summation Example: Use mathematical induction to prove S n = 2 + 4 + 6 + 8 +... + 2n = n(n + 1) for every positive integer n. 1. Show that the formula is true when n = 1. S 1 = n(n + 1) = 1(1 + 1) = 2True 2.Assume the formula is valid for some integer n. 3.Use this assumption to prove the formula S k =k(k + 1) is valid for the next integer, n + 1 and show that the formula S k + 1 = (k + 1)(k + 2) is true. S k = 2 + 4 + 6 + 8 +... + 2k = k(k + 1) Assumption

10. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10 Example(con’t): Use mathematical induction to prove S n = 2 + 4 + 6 + 8 +... + 2n = n(n + 1) for every positive integer n. The formula S n = n(n + 1) is valid for all positive integer values of n. Show that

11. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11 You try: Prove the sum formula: 1) true for n=1? 2) assume is true 3) show that

12. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 12 Sums of Powers of Integers Sums of Powers of Integers :

13. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 13 You Try: Use mathematical induction to prove the following:

14. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 14 Example: Using Induction to Prove Sum of Power Example: Use mathematical induction to prove for all positive integers n, Assumption True Proof:

15. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 15 Example: Using Induction to Prove Sum of Power Example: Use mathematical induction to prove for all positive integers n, Assumption Group terms to form S k. Replace S k by k(k + 1). Example continues. True

16. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 16 Example Continued Simplify. Example continued: The formula is valid for all positive integer values of n.

17. 8.4 Homework Day 1 Pg. 592 7-21 odd, skip #15 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 17

18. 8.4 Mathematical Induction Day 2

19. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 19 Precalculus HWQ 8-4 day 2 Use mathematical induction to prove the formula:

20. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 20 Finite Differences The first differences of the sequence 1, 4, 9, 16, 25, 36 are found by subtracting consecutive terms. n:123456n:123456 a n :149162536 First differences: 357911 Second differences:2222 The second differences are found by subtracting consecutive first differences. quadratic model

21. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 21 Quadratic Models When the second differences are all the same nonzero number, the sequence has a perfect quadratic model. Find the quadratic model for the sequence 1, 4, 9, 16, 25, 36,... a n = an 2 + bn + c a 1 = a(1) 2 + b(1) + c = 1 a 2 = a(2) 2 + b(2) + c = 4 a 3 = a(3) 2 + b(3) + c = 9 Solving the system yields a = 1, b = 0, and c = 0. a n = n 2 a + b + c = 1 4a + 2b + c = 4 9a + 3b + c = 9

22. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 22 Example: Find the Quadratic Model a n = an 2 + bn + c a 0 = a(0) 2 + b(0) + c = 3 a 1 = a(1) 2 + b(1) + c = 3 a 4 = a(4) 2 + b(4) + c = 15 a n = n 2 – n + 3 c = 3 a + 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 a 0 = 3, a 1 = 3, a 4 = 15.

23. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 23 You Try: Take the differences and find a model for the sequence: 5, 8, 11, 14, …

24. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 24 Take the differences and find a model for the sequence. Use the regression feature of your graphing calculator. 3, 5, 8, 12, 17, 23, …

25. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 25 You Try: What type of model fits the data: 2, 13, 46, 113, 226, 397, 638, 961 …

26. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 26 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 …

27. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 27 Recursively defined sequences Write the first 5 terms of the sequence beginning with the given term. Find a model for the sequence.

28. 8.4 Homework Day 2 Pg. 592 8-14 even, 37-43 odd Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 28

29. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 29 Precalculus Warm-Up 2/15 Find a quadratic model for the sequence where:

30. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 30 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.