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Course 3
13-1 Terms of Arithmetic Sequences13-1 Terms of Arithmetic Sequences
Course 3
Warm UpWarm Up
Problem of the DayProblem of the Day
Lesson PresentationLesson Presentation
Course 3
13-1 Terms of 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
Course 3
13-1 Terms of Arithmetic Sequences
17, 15
Course 3
13-1 Terms of 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
Course 3
13-1 Terms of Arithmetic Sequences
Learn to find terms in an arithmetic sequence.
Course 3
13-1 Terms of Arithmetic Sequences
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!
Course 3
13-1 Terms of Arithmetic Sequences
Determine if the sequence could be arithmetic. If so, give the common difference.
5, 8, 11, 14, 17, . . .
Additional Example 1A: Identifying Arithmetic Sequences
The terms increase by 3.
The sequence could be arithmetic with a common difference of 3.
5 8 11 14 17, . . .
3333
Course 3
13-1 Terms of Arithmetic Sequences
Determine if the sequence could be arithmetic. If so, give the common difference.
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
Course 3
13-1 Terms of Arithmetic Sequences
Determine if the sequence could be arithmetic. If so, give the common difference.
65, 60, 55, 50, 45, . . .
Additional Example 1C: Identifying Arithmetic Sequences
The sequence could be arithmetic with a common difference of –5.
The terms decrease by 5.65 60 55 50 45, . . .
–5–5–5–5
Course 3
13-1 Terms of Arithmetic Sequences
Determine if the sequence could be arithmetic. If so, give the common difference.
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.
The terms increase by 0.1.5.7 5.8 5.9 6 6.1, . . .
0.10.10.10.1
Course 3
13-1 Terms of Arithmetic Sequences
Determine if the sequence could be arithmetic. If so, give the common difference.
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
Course 3
13-1 Terms of Arithmetic Sequences
Determine if the sequence could be arithmetic. If so, give the common difference.
1, 2, 3, 4, 5, . . .
Check It Out: Example 1A
The sequence could be arithmetic with a common difference of 1.
The terms increase by 1.1 2 3 4 5, . . .
1111
Course 3
13-1 Terms of Arithmetic Sequences
Determine if the sequence could be arithmetic. If so, give the common difference.
1, 3, 7, 8, 12, …
Check It Out: Example 1B
The sequence is not arithmetic.
Find the difference of each term and the term before it.
1 3 7 8 12, . . .
4142
Course 3
13-1 Terms of Arithmetic Sequences
Determine if the sequence could be arithmetic. If so, give the common difference.
11, 22, 33, 44, 55, . . .
Check It Out: Example 1C
The sequence could be arithmetic with a common difference of 11.
The terms increase by 11.11 22 33 44 55, . . .
11111111
Course 3
13-1 Terms of Arithmetic Sequences
Determine if the sequence could be arithmetic. If so, give the common difference.
1, 1, 1, 1, 1, 1, . . .
Check It Out: 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
Course 3
13-1 Terms of Arithmetic Sequences
Determine if the sequence could be arithmetic. If so, give the common difference.
2, 4, 6, 8, 9, . . .
Check It Out: Example 1E
The sequence is not arithmetic.
Find the difference of each term and the term before it.
2 4 6 8 9, . . .
1222
Course 3
13-1 Terms of Arithmetic Sequences
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.
Course 3
13-1 Terms of Arithmetic Sequences
Find the given term in the arithmetic sequence.
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
Course 3
13-1 Terms of Arithmetic Sequences
Find the given term in the arithmetic sequence.
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
Course 3
13-1 Terms of Arithmetic Sequences
Find the given term in the arithmetic sequence.
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
Course 3
13-1 Terms of Arithmetic Sequences
Find the given term in the arithmetic sequence.
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
Course 3
13-1 Terms of Arithmetic Sequences
Find the given term in the arithmetic sequence.
15th term: 1, 3, 5, 7, . . .
Check it Out: Example 2A
an = a1 + (n – 1)d
a15 = 1 + (15 – 1)2
a15 = 29
Course 3
13-1 Terms of Arithmetic Sequences
Find the given term in the arithmetic sequence.
50th term: 100, 93, 86, 79, . . .
Check It Out: Example 2B
an = a1 + (n – 1)d
a50 = 100 + (50 – 1)(-7)
a50 = –243
Course 3
13-1 Terms of Arithmetic Sequences
Find the given term in the arithmetic sequence.
41st term: 25, 25.5, 26, 26.5, . . .
Check It Out: Example 2C
an = a1 + (n – 1)d
a41 = 25 + (41 – 1)(0.5)
a41 = 45
Course 3
13-1 Terms of Arithmetic Sequences
Find the given term in the arithmetic sequence.
2nd term: a1 = 13, d = 5
Check It Out: Example 2D
an = a1 + (n – 1)d
a2 = 13 + (2 – 1)5
a2 = 18
Course 3
13-1 Terms of Arithmetic Sequences
You can use the formula for the nth term of an arithmetic sequence to solve for other variables.
Course 3
13-1 Terms of 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 a1 = 20.5 = money after first sale
d = 0.5
an = 63.5
d = .50 = common difference
an = 63.5 = money at the end of the sale
Course 3
13-1 Terms of Arithmetic Sequences
Additional Example 3 ContinuedLet n represent the item number of cookies sold that will earn the class a total of $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.
Course 3
13-1 Terms of 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?
Check It Out: Example 3
Identify the arithmetic sequence: 10.25, 10.5, 10.75, 11, …
a1 = 10.25
d = 0.25
an = 40
a1 = 10.25 = money after first sale
d = .25 = common difference
an = 40 = money at the end of the sale
Course 3
13-1 Terms of Arithmetic Sequences
Check It Out: 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.
Course 3
13-1 Terms of 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