1. Terms of Arithmetic Sequences
13-1
Terms of Arithmetic Sequences
Course 3
Warm Up
Problem of the Day
Lesson Presentation

2. Terms of Arithmetic Sequences
Course 3
13-1
Terms of Arithmetic Sequences
Warm Up
Find the next two numbers in the pattern, using the simplest rule you can find.
1. 1, 5, 9, 13, . . .
2. 100, 50, 25, 12.5, . . .
3. 80, 87, 94, 101, . . .
4. 3, 9, 7, 13, 11, . . .
17, 21
6.25, 3.125
108, 115
17, 15

3. Problem of the Day
Write the last part of this set of equations so that its graph is the letter W.
y = –2x + 4 for 0  x  2
y = 2x – 4 for 2 < x  4
y = –2x + 12 for 4 < x  6
Possible answer: y = 2x – 12 for 6 < x  8

4. Learn to find terms in an arithmetic sequence.

5. You cannot tell if a sequence is arithmetic by looking at a finite number of terms because the next term might not fit the pattern.
Caution!

6. Additional Example 1A: Identifying Arithmetic Sequences
Determine if the sequence could be arithmetic. If so, give the common difference.
5, 8, 11, 14, 17, . . .
The terms increase by 3.
, . . . 3 3 3 3
The sequence could be arithmetic with a common difference of 3.

7. Additional Example 1B: Identifying Arithmetic Sequences
Determine if the sequence could be arithmetic. If so, give the common difference.
1, 3, 6, 10, 15, . . .
Find the difference of each term and the term before it.
, . . . 2 3 4 5
The sequence is not arithmetic.

8. Additional Example 1C: Identifying Arithmetic Sequences
Determine if the sequence could be arithmetic. If so, give the common difference.
65, 60, 55, 50, 45, . . .
, . . .
The terms decrease by 5. –5 –5 –5 –5
The sequence could be arithmetic with a common difference of –5.

9. Additional Example 1D: Identifying Arithmetic Sequences
Determine if the sequence could be arithmetic. If so, give the common difference.
5.7, 5.8, 5.9, 6, 6.1, . . .
The terms increase by 0.1.
, . . .
0.1
0.1
0.1
0.1
The sequence could be arithmetic with a common difference of 0.1.

10. Additional Example 1E: Identifying Arithmetic Sequences
Determine if the sequence could be arithmetic. If so, give the common difference.
1, 0, -1, 0, 1, . . .
Find the difference of each term and the term before it.
– , . . . –1 –1 1 1
The sequence is not arithmetic.

11. Check It Out: Example 1A
Determine if the sequence could be arithmetic. If so, give the common difference.
1, 2, 3, 4, 5, . . .
The terms increase by 1.
, . . . 1 1 1 1
The sequence could be arithmetic with a common difference of 1.

12. Check It Out: Example 1B
Determine if the sequence could be arithmetic. If so, give the common difference.
1, 3, 7, 8, 12, …
Find the difference of each term and the term before it.
, . . . 2 4 1 4
The sequence is not arithmetic.

13. Check It Out: Example 1C
Determine if the sequence could be arithmetic. If so, give the common difference.
11, 22, 33, 44, 55, . . .
The terms increase by 11.
, . . . 11 11 11 11
The sequence could be arithmetic with a common difference of 11.

14. Check It Out: Example 1D
Determine if the sequence could be arithmetic. If so, give the common difference.
1, 1, 1, 1, 1, 1, . . .
Find the difference of each term and the term before it.
, . . .
The sequence could be arithmetic with a common difference of 0.

15. Check It Out: Example 1E
Determine if the sequence could be arithmetic. If so, give the common difference.
2, 4, 6, 8, 9, . . .
Find the difference of each term and the term before it.
, . . . 2 2 2 1
The sequence is not arithmetic.

16. Helpful Hint
Subscripts are used to show the positions of terms in the sequence. The first term is a1, “read a sub one,” the second is a2, and so on.

17. Additional Example 2A: Finding a Given Term of an Arithmetic Sequence
Find the given term in the arithmetic sequence.
10th term: 1, 3, 5, 7, . . .
an = a1 + (n – 1)d
a10 = 1 + (10 – 1)2
a10 = 19

18. Additional Example 2B: Finding a Given Term of an Arithmetic Sequence
Find the given term in the arithmetic sequence.
18th term: 100, 93, 86, 79, . . .
an = a1 + (n – 1)d
a18 = (18 – 1)(–7)
a18 = -19

19. Additional Example 2C: Finding a Given Term of an Arithmetic Sequence
Find the given term in the arithmetic sequence.
21st term: 25, 25.5, 26, 26.5, . . .
an = a1 + (n – 1)d
a21 = 25 + (21 – 1)(0.5)
a21 = 35

20. Additional Example 2D: Finding a Given Term of an Arithmetic Sequence
Find the given term in the arithmetic sequence.
14th term: a1 = 13, d = 5
an = a1 + (n – 1)d
a14 = 13 + (14 – 1)5
a14 = 78

21. Check it Out: Example 2A
Find the given term in the arithmetic sequence.
15th term: 1, 3, 5, 7, . . .
an = a1 + (n – 1)d
a15 = 1 + (15 – 1)2
a15 = 29

22. an = a1 + (n – 1)d a50 = 100 + (50 – 1)(-7) a50 = –243
Check It Out: Example 2B
Find the given term in the arithmetic sequence.
50th term: 100, 93, 86, 79, . . .
an = a1 + (n – 1)d
a50 = (50 – 1)(-7)
a50 = –243

23. an = a1 + (n – 1)d a41 = 25 + (41 – 1)(0.5) a41 = 45
Check It Out: Example 2C
Find the given term in the arithmetic sequence.
41st term: 25, 25.5, 26, 26.5, . . .
an = a1 + (n – 1)d
a41 = 25 + (41 – 1)(0.5)
a41 = 45

24. an = a1 + (n – 1)d a2 = 13 + (2 – 1)5 a2 = 18 Check It Out: Example 2D
Find the given term in the arithmetic sequence.
2nd term: a1 = 13, d = 5
an = a1 + (n – 1)d
a2 = 13 + (2 – 1)5
a2 = 18

25. You can use the formula for the nth term of an arithmetic sequence to solve for other variables.

26. Additional Example 3: Application
The senior class held a bake sale. At the beginning of the sale, there was $20 in the cash box. Each item in the sale cost 50 cents. At the end of the sale, there was $63.50 in the cash box. How many items were sold during the bake sale?
Identify the arithmetic sequence: , 21, 21.5, 22, . . .
a1 = 20.5
a1 = 20.5 = money after first sale
d = 0.5
d = .50 = common difference
an = 63.5
an = 63.5 = money at the end of the sale

27. Additional Example 3 Continued
Let n represent the item number of cookies sold that will earn the class a total of $ Use the formula for arithmetic sequences.
an = a1 + (n – 1) d
63.5 = (n – 1)(0.5)
Solve for n.
63.5 = n – 0.5
Distributive Property.
63.5 = n
Combine like terms.
43.5 = 0.5n
Subtract 20 from both sides.
87 = n
Divide both sides by 0.5.
During the bake sale, 87 items are sold in order for the cash box to contain $63.50.

28. Check It Out: Example 3
Johnnie is selling pencils for student council. At the beginning of the day, there was $10 in his money bag. Each pencil costs 25 cents. At the end of the day, he had $40 in his money bag. How many pencils were sold during the day?
Identify the arithmetic sequence: 10.25, 10.5, 10.75, 11, …
a1 = 10.25
a1 = = money after first sale
d = 0.25
d = .25 = common difference
an = 40
an = 40 = money at the end of the sale

29. Check It Out: Example 3 Continued
Let n represent the number of pencils in which he will have $40 in his money bag. Use the formula for arithmetic sequences.
an = a1 + (n – 1)d
40 = (n – 1)(0.25)
Solve for n.
40 = n – 0.25
Distributive Property.
Combine like terms.
40 = n
30 = 0.25n
Subtract 10 from both sides.
120 = n
Divide both sides by 0.25.
120 pencils are sold in order for his money bag to contain $40.

30. Lesson Quiz
Determine if each sequence could be arithmetic. If so, give the common difference.
1. 42, 49, 56, 63, 70, . . .
2. 1, 2, 4, 8, 16, 32, . . .
Find the given term in each arithmetic sequence.
3. 15th term: a1 = 7, d = 5
4. 24th term: 1, , , , 2
5. 52nd term: a1 = 14.2; d = –1.2
yes; 7 no 77 5 4 3 2
, or 6.75 27 4 7 4
–47