CHAPTER 5 Number Theory and the Real Number
CHAPTER 5 Number Theory and the Real Number System © 2010 Pearson Prentice Hall. All rights reserved.
5. 7 Arithmetic and Geometric Sequences © 2010 Pearson Prentice Hall. All rights reserved. 2
1. 2. 3. 4. Objectives Write terms of an arithmetic sequence. Use the formula for the general term of an arithmetic sequence. Write terms of a geometric sequence. Use the formula for the general term of a geometric sequence. © 2010 Pearson Prentice Hall. All rights reserved. 3
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+1=2 © 2010 Pearson Prentice Hall. All rights reserved. 1+2=3 3+2=5 5+3=8 4
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 Sequence Common 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 © 2010 Pearson Prentice Hall. All rights reserved. 5
Example 1: Writing the Terms of an Arithmetic Sequence 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 © 2010 Pearson Prentice Hall. All rights reserved. 6
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 nth term, of an arithmetic sequence: The nth term (general term) of an arithmetic sequence with first term a 1 and common difference d is an = a 1 + (n – 1)d. © 2010 Pearson Prentice Hall. All rights reserved. 7
Example 3: Using the Formula for the General Term of an Arithmetic Sequence 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. an = a 1 + (n – 1)d a 8 = 4 + (8 – 1)( 7) = 4 + 7( 7) = 4 + ( 49) = 45 The eighth term is 45. © 2010 Pearson Prentice Hall. All rights reserved. 8
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 Sequence Common Ratio 1, 5, 25, 125, 625, … 4, 8, 16, 32, 64, … 6, -12, 24, -48, 96, … © 2010 Pearson Prentice Hall. All rights reserved. 9
Example 5: Writing the Terms of a Geometric Sequences 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 © 2010 Pearson Prentice Hall. All rights reserved. 10
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 nth term, of a geometric sequence: The nth term (general term) of a geometric sequence with first term a 1 and common ratio r is an = a 1 r n-1 © 2010 Pearson Prentice Hall. All rights reserved. 11
Example 6: Using the Formula for the General Term of a Geometric Sequence 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. an = a 1 r n-1 a 8 = 4( 2)8 -1 = 4( 2)7 = 4( 128) = 512 The eighth term is 512. © 2010 Pearson Prentice Hall. All rights reserved. 12
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