Download Arithmetic and Geometric Sequences

CHAPTER 5
Number Theory and the
Real Number System
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5.7
Arithmetic and Geometric Sequences
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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.
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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
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1+2=3
3+2=5
5+3=8
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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
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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
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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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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, 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.
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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
1, 5, 25, 125, 625, …
4, 8, 16, 32, 64, …
6, -12, 24, -48, 96, …
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Common Ratio
5
 5
1
8
r 
 2
4
r 
r 
 12
 2
6
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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
2 2 2
2
6,2, , ,
, and
.
3 9 27
81
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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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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, 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.
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