9.2 Arithmetic Sequences and Series 397

Lesson 9.2 Arithmetic Sequences and SeriesA warehouse manager at Wash-A-Lot keeps track of inventory in a logbook. The warehouse ordered several skids of dishwasher detergent. Each skid contains the same number of cartons so that the total number of cartons on 1, 2, 3, 4, … skids follows an arithmetic sequence.

Arithmetic Sequences

An arithmetic sequence is a sequence in which the difference between consecutive terms is the same. The difference between consecutive terms is the common difference of the sequence.

Example 1 Identifying an Arithmetic Sequence

The table at the left shows the bus schedule at a bus stop by Heather’s house. Describe the pattern shown in the table and find the next two pickup times assuming the pattern stays the same.

Solution

Each pickup time is 15 minutes later than the previous time. The bus pickup times represent an arithmetic sequence with a common difference of 15 minutes. The next two pickup times will be 7:20 a.m. and 7:35 a.m.

Ongoing AssessmentDetermine if the sequence 0.4, 1.0, 1.6, 2.2, 2.8, 3.4, … is arithmetic. If so, state the common difference and find the next three terms of the sequence.

ObjectivesIdentify an arithmetic sequence.Write and evaluate an arithmetic series.

Bus SchedulePickup

Number Time

1 6:05 a.m.2 6:20 a.m.3 6:35 a.m.4 6:50 a.m.5 7:05 a.m.

398 Chapter 9 Sequences and Series

The arithmetic mean of two numbers is the average of the two numbers.

arithmetic mean = sum of two numbers2

The arithmetic mean can be used to find a missing term in an arithmetic sequence.

Example 2 Finding an Arithmetic Mean

In the introductory paragraph of this lesson, it was stated that each skid contains the same number of cartons. Therefore, the total number of cartons on 1, 2, 3, 4, … skids follows an arithmetic sequence.

Suppose the manager writes the arithmetic sequence …, 105, ?, 135, … . Find the missing term of the arithmetic sequence to complete the manager’s logbook.

Solution

Use the arithmetic mean to find the missing term of the arithmetic sequence.

arithmetic mean = 105 135+2

= 120

The missing number from the logbook is 120 cartons of detergent.

Critical Thinking Explain how to find two consecutive missing terms from an arithmetic sequence. Then find the missing terms in the sequence …, 11, ?, ?, 32, … .

Arithmetic Series

A series is a mathematical expression for the sum of the terms of a sequence. A series can have a finite number of terms that can be counted individually such as 1 + 2 + 3 + 4, or it can have an infinite number of terms such as 2 + 4 + 6 + 8 + 10 + … . An arithmetic series is a series whose terms form an arithmetic sequence.

Sum of an Arithmetic Series

The sum Sn of an arithmetic series with a finite number of terms, such as a1 + a2 + a3 + … + an, is given by the formula

Sn = n2

(a1 + an)

where a1 is the first term, an is the last term, and n is the number of terms in the series.

9.2 Arithmetic Sequences and Series 399

Summation Notation

Summation notation is used to write the sum of the terms of a series in a compact way. The Greek letter sigma, ∑, is used in this notation. Under the sigma symbol is the lower limit of the series and an index indicated by n = 1. Above the symbol is the upper limit, which is the greatest value of n. To the right of the symbol is an explicit formula for the terms of the sequence.

31

5

nn=∑ = 3(1) + 3(2) + 3(3) + 3(4) + 3(5) = 45

Example 3 Writing and Evaluating an Arithmetic Series

A school auditorium has 20 rows of seats. The first row has 14 seats, and each successive row has 2 more seats than the previous row. Write and evaluate an arithmetic series to find the total number of seats in the auditorium. Use summation notation.

Solution

Write an expression for the number of seats in the nth row.

Row 1: 14 + 0(2) = 14 Row 2: 14 + 1(2) = 16 Row 3: 14 + 2(2) = 18 Row n: 14 + (n – 1)(2) = 12 + 2n

Write the series in summation notation.

12 121

20

+=∑ nn

Find the first and last terms of the arithmetic series.

a1 = 12 + 2(1) = 14 a20 = 12 + 2(20) = 52

Use the formula for the sum of a finite arithmetic series.

Sn = 202

(14 + 52) = 660

There are 660 seats in the school auditorium.

400 Chapter 9 Sequences and Series

Ongoing Assessment

Find the first and last terms of the series 4 11

16

( )nn

+=∑ . Then evaluate

the series.

Lesson AssessmentThink and Discuss

1. How can the common difference of an arithmetic sequence be found?

2. Explain how to find the sum of the first 100 positive integers.

3. Describe the difference between an arithmetic sequence and an arithmetic series.

4. Describe the arithmetic mean of two numbers in your own words.

5. Is it possible for the common difference of an arithmetic sequence to be a negative number? Explain.

Practice and Problem Solving

Determine if each sequence is arithmetic. If so, state the common difference of the sequence.

6. 3, 9, 15, 21, 27, … 7. 3, 2, 1, 0, –1, …

8. 13

231 43

53

, , , , ,… 13

9. 8.2, 8.5, 8.8, 9.1, 9.4, …

10. 5, 7, 9, 12, 15, … 11. 0, –0.1, –0.2, –0.3, –0.4, …

12. 1, 4, 9, 16, 25, … 13. –7, –3, 1, 5, 9, …

Use the arithmetic mean to find the missing term in each arithmetic sequence.

14. 6, 13, 20, , 34, …

15. –2.2, –1.9, , –1.3, –1, …

16. 246, , 500, 627, 754, …

17. 317, 215, , 11, –91, …

9.2 Arithmetic Sequences and Series 401

Use the formula Sn = n2 (a1 + an) to evaluate each series.

18. 31

50

nn=∑ 19. 2 1

1

82

( )nn

−=∑

20. 3 21

34

( )nn

+=∑ 21. 7 5

1

62

nn

−=∑

22. 8 11

53

nn

−=∑ 23. 5 2

1

25

nn

+=∑

24. A marching band has 1 member in the front row, 2 members in the second row, 3 members in the third row, and so on. If there are 20 rows altogether, how many band members are there?

25. Jeremy used square tiles to create a pattern. His pattern is shown below. Does the pattern represent an arithmetic sequence? If so, state the common difference.

26. A quality control engineer sampled certain items as they came down an assembly line to check for functionality. The numbers followed the pattern 17, 43, 69, , 121, … . Use the arithmetic mean to find the missing term in the sequence.

27. The pattern $20, $24.50, $28.50, $32.50, $36.50, … describes the amount of money in Kim’s savings account over several days. Does the pattern represent an arithmetic sequence? If so, state the common difference.

402 Chapter 9 Sequences and Series

28. The table below shows the times of the first five tours at an art museum. Do the times represent an arithmetic sequence? If so, state the common difference.

Tour Schedule

Tour Number Time

1 8:15 a.m.

2 9:35 a.m.

3 10:55 a.m.

4 12:15 p.m.

5 1:35 p.m.

29. A prospector found 0.5 ounce of gold on the first day, 0.85 ounce on the second day, 1.2 ounces on the third day, 1.55 ounces on the fourth day, and so on. If this pattern continued for 14 days, how many ounces of gold did the prospector find altogether?

30. Mr. Williams turned on a hose to fill a swimming pool. After 12 minutes, the water was 8 inches deep. He wanted to turn the water off when it reached a height of 1.5 feet. How much longer did it take to fill the pool to a height of 1.5 feet?

31. A football team gained 2 yards, 3 yards, 5 yards, 8 yards, and 12 yards on their first five plays of the game. Do these numbers represent an arithmetic sequence? If so, state the common difference.

Mixed Review

Divide using synthetic division.

32. (h3 + h2 + 5h + 5) ÷ (h + 1)

33. (2r3 – 7r2 + 5r + 2) ÷ (r – 2)

34. (3x3 – 9x2 + 5x – 15) ÷ (x – 3)