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5.7

Arithmetic and Geometric Sequences

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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.

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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+2=3 3+2=5 5+3=81+1=2

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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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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

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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