Pre-Algebra 12-1 Arithmetic Sequences Pre-Algebra HOMEWORK Page 606 #1-9
Pre-Algebra
12-1 Arithmetic Sequences
Pre-Algebra HOMEWORK
Page 606
#1-9
Pre-Algebra
12-1 Arithmetic Sequences
Students will be able to solve sequences and represent functions by completing the following assignments.
• Learn to find terms in an arithmetic sequence.• Learn to find terms in a geometric sequence.• Learn to find patterns in sequences.• Learn to represent functions with tables, graphs, or equations.
Pre-Algebra
12-1 Arithmetic Sequences
Today’s Learning Goal Assignment
Learn to find terms in an arithmetic sequence.
Pre-Algebra
12-1 Arithmetic Sequences12-1 Arithmetic Sequences
Pre-Algebra
Warm UpWarm Up
Problem of the DayProblem of the Day
Lesson PresentationLesson Presentation
Pre-Algebra
12-1 Arithmetic Sequences
Warm UpFind 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
Pre-Algebra
12-1 Arithmetic Sequences
17, 15
Pre-Algebra
12-1 Arithmetic Sequences
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 2y = 2x – 4 for 2 < x 4y = –2x + 12 for 4 < x 6
Possible answer: y = 2x – 12 for 6 < x 8
Pre-Algebra
12-1 Arithmetic Sequences
Vocabulary
sequencetermarithmetic sequencecommon difference
Pre-Algebra
12-1 Arithmetic Sequences
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.
Pre-Algebra
12-1 Arithmetic Sequences
Determine if the sequence could be arithmetic. If so, give the common difference.
A. 5, 8, 11, 14, 17, . . .
Additional Example 1A: Identifying Arithmetic Sequences
Find the difference of each term and the term before it.
The sequence could be arithmetic with a common difference of 3.
5 8 11 14 17, . . .
3333
Pre-Algebra
12-1 Arithmetic Sequences
Determine if the sequence could be arithmetic. If so, give the common difference.
A. 1, 2, 3, 4, 5, . . .
Try This: Example 1A
The sequence could be arithmetic with a common difference of 1.
Find the difference of each term and the term before it.
1 2 3 4 5, . . .
1111
Pre-Algebra
12-1 Arithmetic Sequences
Determine if the sequence could be arithmetic. If so, give the common difference.
B. 1, 3, 6, 10, 15, . . .
Additional Example 1B: Identifying Arithmetic Sequences
The sequence is not arithmetic.
Find the difference of each term and the term before it.
1 3 6 10 15, . . .
5432
Pre-Algebra
12-1 Arithmetic Sequences
Determine if the sequence could be arithmetic. If so, give the common difference.
B. 1, 3, 7, 8, 12, …
Try This: Example 1B
The sequence is not arithmetic.
Find the difference of each term and the term before it.
1 3 7 8 12, . . .
4142
Pre-Algebra
12-1 Arithmetic Sequences
Determine if the sequence could be arithmetic. If so, give the common difference.
C. 65, 60, 55, 50, 45, . . .
Additional Example 1C: Identifying Arithmetic Sequences
The sequence could be arithmetic with a common difference of –5.
Find the difference of each term and the term before it.
65 60 55 50 45, . . .
–5–5–5–5
Pre-Algebra
12-1 Arithmetic Sequences
Determine if the sequence could be arithmetic. If so, give the common difference.
C. 11, 22, 33, 44, 55, . . .
Try This: Example 1C
The sequence could be arithmetic with a common difference of 11.
Find the difference of each term and the term before it.
11 22 33 44 55, . . .
11111111
Pre-Algebra
12-1 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, . . .
Additional Example 1D: Identifying Arithmetic Sequences
The sequence could be arithmetic with a common difference of 0.1.
Find the difference of each term and the term before it.
5.7 5.8 5.9 6 6.1, . . .
0.10.10.10.1
Pre-Algebra
12-1 Arithmetic Sequences
Determine if the sequence could be arithmetic. If so, give the common difference.
D. 1, 1, 1, 1, 1, 1, . . .
Try This: Example 1D
The sequence could be arithmetic with a common difference of 0.
Find the difference of each term and the term before it.
1 1 1 1 1, . . .
0000
Pre-Algebra
12-1 Arithmetic Sequences
Determine if the sequence could be arithmetic. If so, give the common difference.
E. 1, 0, -1, 0, 1, . . .
Additional Example 1E: Identifying Arithmetic Sequences
The sequence is not arithmetic.
Find the difference of each term and the term before it.
1 0 –1 0 1, . . .
11–1–1
Pre-Algebra
12-1 Arithmetic Sequences
Determine if the sequence could be arithmetic. If so, give the common difference.
E. 2, 4, 6, 8, 9, . . .
Try This: Example 1E
The sequence is not arithmetic.
Find the difference of each term and the term before it.
2 4 6 8 9, . . .
1222
Pre-Algebra
12-1 Arithmetic Sequences
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.
Pre-Algebra
12-1 Arithmetic Sequences
Find the given term in the arithmetic sequence.
A. 10th term: 1, 3, 5, 7, . . .
Additional Example 2A: Finding a Given Term of an Arithmetic Sequence
an = a1 + (n – 1)d
a10 = 1 + (10 – 1)2
a10 = 19
Pre-Algebra
12-1 Arithmetic Sequences
Find the given term in the arithmetic sequence.
A. 15th term: 1, 3, 5, 7, . . .
Try This: Example 2A
an = a1 + (n – 1)d
a15 = 1 + (15 – 1)2
a15 = 29
Pre-Algebra
12-1 Arithmetic Sequences
Find the given term in the arithmetic sequence.
B. 18th term: 100, 93, 86, 79, . . .
Additional Example 2B: Finding a Given Term of an Arithmetic Sequence
an = a1 + (n – 1)d
a18 = 100 + (18 – 1)(–7)
a18 = -19
Pre-Algebra
12-1 Arithmetic Sequences
Find the given term in the arithmetic sequence.
B. 50th term: 100, 93, 86, 79, . . .
Try This: Example 2B
an = a1 + (n – 1)d
a50 = 100 + (50 – 1)(-7)
a50 = –243
Pre-Algebra
12-1 Arithmetic Sequences
Find the given term in the arithmetic sequence.
C. 21st term: 25, 25.5, 26, 26.5, . . .
Additional Example 2C: Finding a Given Term of an Arithmetic Sequence
an = a1 + (n – 1)d
a21 = 25 + (21 – 1)(0.5)
a21 = 35
Pre-Algebra
12-1 Arithmetic Sequences
Find the given term in the arithmetic sequence.
C. 41st term: 25, 25.5, 26, 26.5, . . .
Try This: Example 2C
an = a1 + (n – 1)d
a41 = 25 + (41 – 1)(0.5)
a41 = 45
Pre-Algebra
12-1 Arithmetic Sequences
Find the given term in the arithmetic sequence.
D. 14th term: a1 = 13, d = 5
Additional Example 2D: Finding a Given Term of an Arithmetic Sequence
an = a1 + (n – 1)d
a14 = 13 + (14 – 1)5
a14 = 78
Pre-Algebra
12-1 Arithmetic Sequences
Find the given term in the arithmetic sequence.
D. 2nd term: a1 = 13, d = 5
Try This: Example 2D
an = a1 + (n – 1)d
a2 = 13 + (2 – 1)5
a2 = 18
Pre-Algebra
12-1 Arithmetic Sequences
You can use the formula for the nth term of an arithmetic sequence to solve for other variables.
Pre-Algebra
12-1 Arithmetic Sequences
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?
Additional Example 3: Application
Identify the arithmetic sequence: 20.5, 21, 21.5, 22, . . .
a1 = 20.5 Let a1 = 20.5 = money after first sale.
d = 0.5
an = 63.5
Pre-Algebra
12-1 Arithmetic Sequences
Additional Example 3 Continued
Let n represent the item number in which the cash box will contain $63.50. Use the formula for arithmetic sequences.
an = a1 + (n – 1) d
Solve for n.63.5 = 20.5 + (n – 1)(0.5)
63.5 = 20.5 + 0.5n – 0.5 Distributive Property.
63.5 = 20 + 0.5n Combine like terms.
87 = n
Subtract 20 from both sides.
During the bake sale, 87 items are sold in order for the cash box to contain $63.50.
43.5 = 0.5n
Divide both sides by 0.5.
Pre-Algebra
12-1 Arithmetic Sequences
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?
Try This: Example 3
Identify the arithmetic sequence: 10.25, 10.5, 10.75, 11, …
a1 = 10.25 Let a1 = 10.25 = money after first sale.
d = 0.25
an = 40
Pre-Algebra
12-1 Arithmetic Sequences
Try This: Example 3 ContinuedLet 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
Solve for n.40 = 10.25 + (n – 1)(0.25)
40 = 10.25 + 0.25n – 0.25 Distributive Property.
40 = 10 + 0.25n Combine like terms.
120 = n
Subtract 10 from both sides.
120 pencils are sold in order for his money bag to contain $40.
30 = 0.25n
Divide both sides by 0.25.
Pre-Algebra
12-1 Arithmetic Sequences
Lesson QuizDetermine 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
no
yes; 7
77
54
32
74
, or 6.7527 4
–47