Copyright © 2011 Pearson Education, Inc. Slide A sequence in which each term after the first is obtained by adding a fixed number to the previous term is an arithmetic sequence (or arithmetic progression). The fixed number that is added is the common difference. 5, 9, 13, 17 … is an example of an arithmetic sequence since 4 is added to each term to get the next term Arithmetic Sequences and Series

Copyright © 2011 Pearson Education, Inc. Slide Finding a Common Difference Example Find the common difference d for the arithmetic sequence –9, –7, –5, –3, –1, … Solution d can be found by choosing any two consecutive terms and subtracting the first from the second: d = –5 – (–7) = 2.

Copyright © 2011 Pearson Education, Inc. Slide Arithmetic Sequences and Series nth Term of an Arithmetic Sequence In an arithmetic sequence with first term a 1 and common difference d, the nth term is given by

Copyright © 2011 Pearson Education, Inc. Slide Finding Terms of an Arithmetic Sequence Example Find a 13 and a n for the arithmetic sequence –3, 1, 5, 9, … Solution Here a 1 = –3 and d = 1 – (–3) = 4. Using n=13, In general

Copyright © 2011 Pearson Education, Inc. Slide Your turn. Find a 7 and a n for the arithmetic sequence 7, 10, 13, 16,….

Copyright © 2011 Pearson Education, Inc. Slide Find the nth term from a Graph Example Find a formula for the nth term of the sequence graphed below.

Copyright © 2011 Pearson Education, Inc. Slide Find the nth term from a Graph Solution The equation of the dashed line shown Below is y = –0.5x +4. The sequence is given by a n = –0.5n +4 for n = 1, 2, 3, 4, 5, 6.

Copyright © 2011 Pearson Education, Inc. Slide Work with a Partner/Group Find the sum off all of the integers from 1 to 100 (inclusive). Think of ways to minimize your workload. Be prepared to share your ideas with the class.

Copyright © 2011 Pearson Education, Inc. Slide Arithmetic Sequences and Series Sum of the First n Terms of an Arithmetic Sequence If an arithmetic sequence has first term a 1 and common difference d, the sum of the first n terms is given by or

Copyright © 2011 Pearson Education, Inc. Slide Using The Sum Formulas Example Find the sum of the first 60 positive integers. Solution The sequence is 1, 2, 3, …, 60 so a 1 = 1 and a 60 = 60. The desired sum is

Copyright © 2011 Pearson Education, Inc. Slide Using Summation Notation Example Evaluate the sum. Solution The sum contains the terms of an arithmetic sequence having a 1 = 4(1) + 8 = 12 and a 10 = 4(10) + 8 = 48. Thus,

Copyright © 2011 Pearson Education, Inc. Slide More Practice 1.a 15 = 8, a 17 = 2, determine a 10 and a n. 2.Determine the sum of the first 5 terms of the arithmetic sequence. a 3 = 5, a 4 = 8. 3.