1. Arithmetic Sequences and Series 2015 2 days Digital Lesson

2. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 Definition of Sequence 1/26/2015 Precalculus HWQ: Simplify the factorial expression: !!!!

3. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 3 Definition of Sequence Precalculus Warm-up Write an expression for the apparent nth term of the sequence:

4. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4 An infinite sequence is a function whose domain is the set of positive integers. a 1, a 2, a 3, a 4,..., a n,... Definition of Sequence The first three terms of the sequence a n = 4n – 7 are a 1 = 4(1) – 7 = – 3 a 2 = 4(2) – 7 = 1 a 3 = 4(3) – 7 = 5. finite sequence terms

5. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5 A sequence is arithmetic if the differences between consecutive terms are the same. 4, 9, 14, 19, 24,... Definition of Arithmetic Sequence 9 – 4 = 5 14 – 9 = 5 19 – 14 = 5 24 – 19 = 5 arithmetic sequence The common difference, d, is 5.

6. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 6 Example: Find the first five terms of the sequence and determine if it is arithmetic. a n = 1 + (n – 1)4 Example: Arithmetic Sequence This is an arithmetic sequence. d = 4 a 1 = 1 + (1 – 1)4 = 1 + 0 = 1 a 2 = 1 + (2 – 1)4 = 1 + 4 = 5 a 3 = 1 + (3 – 1)4 = 1 + 8 = 9 a 4 = 1 + (4 – 1)4 = 1 + 12 = 13 a 5 = 1 + (5 – 1)4 = 1 + 16 = 17

7. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7 Determine whether or not each sequence is arithmetic. a) -12, -7, -2, 3, 8,... Definition of Arithmetic Sequence b) ln1, ln2, ln3, ln4, ln5,...

8. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8 The nth term of an arithmetic sequence has the form a n = a 1 + (n – 1)d or the alternate form: a n = dn + c where d is the common difference and c = a 1 – d. The nth Term of an Arithmetic Sequence 2, 8, 14, 20, 26,.... d = 8 – 2 = 6 a 1 = 2 c = 2 – 6 = – 4 The nth term is 6n – 4.

9. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9 a 1 – d = Example: Formula for the nth Term Example: Find the formula for the nth term of an arithmetic sequence whose common difference is 4 and whose first term is 15. Find the first five terms of the sequence. a n = dn + c = 4n + 11 15, d = 4 a 1 = 15 19, 23, 27, 31. The first five terms are 15 – 4 = 11

10. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10 Graphing Utility: Terms and Sum of a Sequence Graphing Utility: Find the first 5 terms of the arithmetic sequence a n = 4n + 11. List Menu: variable beginning value end value

11. Example: Find the formula for the nth term of an arithmetic sequence whose 4th term is 18 and whose 13th term is 63. Find the 20th term of the sequence. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11

12. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 12 You Try: Find the formula for the nth term of an arithmetic sequence whose 10th term is 32 and whose 16th term is 50. Find the 30th term of the sequence.

13. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 13 Try another: Find the formula for the nth term of an arithmetic sequence whose 5th term is 190 and whose 10th term is 115. Find the 15th term of the sequence.

14. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 14 Try another: Find the formula for the nth term of an arithmetic sequence whose 10th term is -330 and whose 20th term is -450. Find the 52nd term of the sequence.

15. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 15 Ex: Find the sum:

16. Homework Day 1 Pg. 573 1-41 odds only Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 16

17. Arithmetic Sequences and Series Day 2 2015 Digital Lesson

18. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 18 Definition of Sequence 1/27/2015 Precalculus Warm-up : Find a formula for the arithmetic sequence where

19. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 19 Definition of Sequence Precalculus Warm-up: Simplify the factorial expression:

20. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 20 The sum of the first n terms of a sequence is represented by summation notation. Definition of Summation Notation index of summation upper limit of summation lower limit of summation

21. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 21 Definition of Series Consider the infinite sequence a 1, a 2, a 3,..., a i,.... 1. The sum of the first n terms of the sequence is called a finite series or the partial sum of the sequence. a 1 + a 2 + a 3 +... + a n 2.The sum of all the terms of the infinite sequence is called an infinite series. a 1 + a 2 + a 3 +... + a i +...

22. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 22 Example: Find the sum:

23. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 23 The Sum of a Finite Arithmetic Sequence The sum of a finite arithmetic sequence with n terms is given by 5 + 10 + 15 + 20 + 25 + 30 + 35 + 40 + 45 + 50 = ? n = 10 a 1 = 5a 10 = 50

24. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 24 Example: The nth Partial Sum The sum of the first n terms of an infinite sequence is called the nth partial sum. a n = dn + c = 4n – 10 Example: Find the 50th partial sum of the arithmetic sequence – 6, – 2, 2, 6,... a 50 = 4(50) – 10 = 190

25. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 25 Example: Summation Notation Example: Find the partial sum. a1a1 a 100

26. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 26 In an arithmetic sequence, the 20 th term is 116 and the 24 th term is 140. Find the sum of the first 50 terms.

27. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 27 In an arithmetic sequence, the 12 th term is 25 and the 30 th term is 97. Find the sum of the first 40 terms.

28. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 28 Graphing Utility: Terms and Sum of a Sequence Graphing Utility: Find the first 5 terms of the arithmetic sequence a n = 4n + 11. List Menu: variable beginning value end value Graphing Utility: Find the sum List Menu: lower limit upper limit

29. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 29 Example: Find the 150 th partial sum of the sequence: 5, 16, 27, 38, 49, …

30. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 30 A stadium has 20 rows of seats. There are 20 seats in row 1, 21 in row 2, 22 in row 3, etc. How many total seats are there?

31. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 31 Find a formula to represent the sum of n positive odd integers.

32. Homework Day 2 Pg. 573 43-81 odds only Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 32

33. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 33 Example: Sum of Partial Series 1/27/2015 Precalculus HWQ: Find the fourth partial sum of