© 2010 Pearson Prentice Hall. All rights reserved. 1 5.7 Arithmetic and Geometric Sequences.
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Transcript of © 2010 Pearson Prentice Hall. All rights reserved. 1 5.7 Arithmetic and Geometric Sequences.
© 2010 Pearson Prentice Hall. All rights reserved. 2
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.
© 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+2=3 3+2=5 5+3=81+1=2
© 2010 Pearson Prentice Hall. All rights reserved. 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
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
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The General Term of an Arithmetic Sequence
• Consider an arithmetic sequence with first term a1. 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 a1 and common difference d is
an = a1 + (n – 1)d.
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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, a8, we replace n in the formula with 8, a1 with 4, and d with 7.
an = a1 + (n – 1)d
a8 = 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
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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, …
51
5r
24
8r
26
12
r
© 2010 Pearson Prentice Hall. All rights reserved. 9
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
.81
2 and ,
27
2,
9
2,
3
2,2,6
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The General Term of a Geometric Sequence
• Consider a geometric sequence with first term a1 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 a1 and common ratio r is
an = a1r n-1
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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, a8, we replace n in the formula with 8, a1 with 4, and r with 2.
an = a1r n-1
a8 = 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