1. 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 45
Bus 2
Ahmedi Abdulkahar
Arango Laura
Blackman Tia
Bohn Sebastian
Burkett Jayson
Cassisa Dominic
Castellano Joe
Coban John
Correggio Joe
Da Motta Livia
Dasgupta Riyana
Dionne Danielle
Duque Michelle
Gerzina Lindsey
Gordon Taylor
Gordon Macie
Kibler Steven
Lara Ariana
Long Shuangxou
Martins Yves
Mendez Andres
Midyette Angela
Miller Zachary
Narpiel Brandon
Nathan Jordan
Nelson Shannon
Ollestad Stefan
Orlowski Cristina
Pollack Leah
Pomerantz Jessica
Rhodes Adam
Roth Bailey
Rucco Dominic
Salamanca Oscar
Secatello Amanda
Serafini Nicole
Silver Oliver
Sincoff Danielle
Towbin Joey
Vela Melissa
Walker Kirsten
Webb Katarina
Wiernicki Richard
Woods Ashley
Zhou Kelly
108, 115
17, 15

2. Arithmetic Sequences
12.1
Pre-Algebra

3. Learn to find terms in an arithmetic sequence.

4. Vocabulary
sequence
term
arithmetic sequence
common difference

5. A sequence is a list of numbers or objects, called terms, in a certain order. In an arithmetic sequence, the difference between one term and the next is always the same. This difference is called the common difference. The common difference is added to each term to get the next term.

6. Example: Identifying Arithmetic Sequences
Determine if the sequence could be arithmetic. If so, give the common difference.
A. 5, 8, 11, 14, 17, . . .
Find the difference of each term and the term before it.
, . . . 3 3 3 3
The sequence could be arithmetic with a common difference of 3.

7. Example: Identifying Arithmetic Sequences
Determine if the sequence could be arithmetic. If so, give the common difference.
B. 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. Example: Identifying Arithmetic Sequences
Determine if the sequence could be arithmetic. If so, give the common difference.
C. 65, 60, 55, 50, 45, . . .
Find the difference of each term and the term before it.
, . . . –5 –5 –5 –5
The sequence could be arithmetic with a common difference of –5.

9. Example: Identifying Arithmetic Sequences
Determine if the sequence could be arithmetic. If so, give the common difference.
D. 5.7, 5.8, 5.9, 6, 6.1, . . .
Find the difference of each term and the term before it.
, . . .
0.1
0.1
0.1
0.1
The sequence could be arithmetic with a common difference of 0.1.

10. Example: Identifying Arithmetic Sequences
Determine if the sequence could be arithmetic. If so, give the common difference.
E. 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. Try This
Determine if the sequence could be arithmetic. If so, give the common difference.
A. 1, 2, 3, 4, 5, . . .
Find the difference of each term and the term before it.
, . . . 1 1 1 1
The sequence could be arithmetic with a common difference of 1.

12. Try This
Determine if the sequence could be arithmetic. If so, give the common difference.
B. 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. Try This
Determine if the sequence could be arithmetic. If so, give the common difference.
C. 11, 22, 33, 44, 55, . . .
Find the difference of each term and the term before it.
, . . . 11 11 11 11
The sequence could be arithmetic with a common difference of 11.

14. Try This
Determine if the sequence could be arithmetic. If so, give the common difference.
D. 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. Try This
Determine if the sequence could be arithmetic. If so, give the common difference.
E. 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. The nth term an of an arithmetic sequence with common difference d is
Writing Math
Subscripts are used to show the positions of terms in the sequence. The first term is a1, the second is a2, and so on.
FINDING THE nth TERM OF AN ARITHMETIC SEQUENCE
The nth term an of an arithmetic sequence with common difference d is
an = a1 + (n – 1)d.
The term we are looking for.
The FIRST term in the series.
The NUMBER of the term we are looking for.
The COMMON DIFFERENCE.

17. Example an = a1 + (n – 1)d a10 = 1 + (10 – 1)2 a10 = 19
Find the given term in the arithmetic sequence.
A. 10th term: 1, 3, 5, 7, . . .
an = a1 + (n – 1)d
a10 = 1 + (10 – 1)2
a10 = 19

18. Example an = a1 + (n – 1)d a18 = 100 + (18 – 1)(–7) a18 = -19
Find the given term in the arithmetic sequence.
B. 18th term: 100, 93, 86, 79, . . .
an = a1 + (n – 1)d
a18 = (18 – 1)(–7)
a18 = -19

19. Example an = a1 + (n – 1)d a21 = 25 + (21 – 1)(0.5) a21 = 35
Find the given term in the arithmetic sequence.
C. 21st term: 25, 25.5, 26, 26.5, . . .
an = a1 + (n – 1)d
a21 = 25 + (21 – 1)(0.5)
a21 = 35

20. Example an = a1 + (n – 1)d a14 = 13 + (14 – 1)5 a14 = 78
Find the given term in the arithmetic sequence.
D. 14th term: a1 = 13, d = 5
an = a1 + (n – 1)d
a14 = 13 + (14 – 1)5
a14 = 78

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

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

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

24. Try This an = a1 + (n – 1)d a2 = 13 + (2 – 1)5 a2 = 18
Find the given term in the arithmetic sequence.
D. 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. Example
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
Let a1 = 20.5 = money after first sale.
d = 0.5
an = 63.5

27. Example
Let n represent the item number in which the cash box will contain $ 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. Try This
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
Let a1 = = money after first sale.
d = 0.25
an = 40

29. Try This
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