1. © 2010 Pearson Prentice Hall. All rights reserved. CHAPTER 5 Number Theory and the Real Number System

2. © 2010 Pearson Prentice Hall. All rights reserved. 2 5.7 Arithmetic and Geometric Sequences

3. © 2010 Pearson Prentice Hall. All rights reserved. 3 Objectives 1.Write terms of an arithmetic sequence. 2.Use the formula for the general term of an arithmetic sequence. 3.Write terms of a geometric sequence. 4.Use the formula for the general term of a geometric sequence.

4. © 2010 Pearson Prentice Hall. All rights reserved. 4 Sequences A sequence is a list of numbers that are related to each other by a rule. The numbers in the sequence are called its terms. For example, a Fibonacci sequence term takes the sum of the two previous successive terms, i.e., 1+2=33+2=55+3=81+1=2

5. © 2010 Pearson Prentice Hall. All rights reserved. 5 Arithmetic Sequences An arithmetic sequence is a sequence in which each term after the first differs from the preceding term by a constant amount. The difference between consecutive terms is called the common difference of the sequence. Arithmetic SequenceCommon Difference 142, 146, 150, 154, 158, …d = 146 – 142 = 4 -5, -2, 1, 4, 7, …d = -2 – (-5) = -2 + 5 = 3 8, 3, -2, -7, -12, …d = 3 – 8 = -5

6. © 2010 Pearson Prentice Hall. All rights reserved. 6 Write the first six terms of the arithmetic sequence with first term 6 and common difference 4. Solution: The first term is 6. The second term is 6 + 4 = 10. The third term is 10 + 4 = 14, and so on. The first six terms are 6, 10, 14, 18, 22, and 26 Example 1: Writing the Terms of an Arithmetic Sequence

7. © 2010 Pearson Prentice Hall. All rights reserved. 7 The General Term of an Arithmetic Sequence Consider an arithmetic sequence with first term a 1. Then the first six terms are Using the pattern of the terms results in the following formula for the general term, or the n th term, of an arithmetic sequence: The nth term (general term) of an arithmetic sequence with first term a 1 and common difference d is a n = a 1 + (n – 1)d.

8. © 2010 Pearson Prentice Hall. All rights reserved. 8 Find the eighth term of the arithmetic sequence whose first term is 4 and whose common difference is  7. Solution: To find the eighth term, a 8, we replace n in the formula with 8, a 1 with 4, and d with  7. a n = a 1 + (n – 1)d a 8 = 4 + (8 – 1)(  7) = 4 + 7(  7) = 4 + (  49) =  45 The eighth term is  45. Example 3: Using the Formula for the General Term of an Arithmetic Sequence

9. © 2010 Pearson Prentice Hall. All rights reserved. 9 Geometric Sequences A geometric sequence is a sequence in which each term after the first is obtained by multiplying the preceding term by a fixed nonzero constant. The amount by which we multiply each time is called the common ratio of the sequence. Geometric SequenceCommon Ratio 1, 5, 25, 125, 625, … 4, 8, 16, 32, 64, … 6, -12, 24, -48, 96, …

10. © 2010 Pearson Prentice Hall. All rights reserved. 10 Write the first six terms of the geometric sequence with first term 6 and common ratio ⅓. Solution: The first term is 6. The second term is 6 · ⅓ = 2. The third term is 2 · ⅓ = ⅔, and so on. The first six terms are Example 5: Writing the Terms of a Geometric Sequences

11. © 2010 Pearson Prentice Hall. All rights reserved. 11 The General Term of a Geometric Sequence Consider a geometric sequence with first term a 1 and common ratio r. Then the first six terms are Using the pattern of the terms results in the following formula for the general term, or the n th term, of a geometric sequence: The nth term (general term) of a geometric sequence with first term a 1 and common ratio r is a n = a 1 r n-1

12. © 2010 Pearson Prentice Hall. All rights reserved. 12 Find the eighth term in the geometric sequence whose first term is  4 and whose common ratio is  2. Solution: To find the eighth term, a 8, we replace n in the formula with 8, a 1 with  4, and r with  2. a n = a 1 r n-1 a 8 =  4(  2) 8-1 =  4(  2) 7 =  4(  128) = 512 The eighth term is 512. Example 6: Using the Formula for the General Term of a Geometric Sequence