Transcript Slide 1

Warm Up
Find the next two numbers in the pattern,
using the simplest rule you can find.
1. 1, 5, 9, 13, . . . 17, 21
2. 100, 50, 25, 12.5, . . . 6.25, 3.125
3. 80, 87, 94, 101, . . . 108, 115
4. 3, 9, 7, 13, 11,
...
17, 15
Arithmetic Sequences
12.1
Pre-Algebra
Learn to find terms in an arithmetic
sequence.
Vocabulary
sequence
term
arithmetic sequence
common difference
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.
Example: Identifying Arithmetic Sequences
Determine if the sequence could be arithmetic.
If so, give the common difference.
A. 5, 8, 11, 14, 17, . . .
5
8
3
11
3
14
3
17, . . . Find the difference of each
term and the term before
it.
3
The sequence could be arithmetic with a common
difference of 3.
Example: Identifying Arithmetic Sequences
Determine if the sequence could be arithmetic.
If so, give the common difference.
B. 1, 3, 6, 10, 15, . . .
1
3
2
6
3
10
4
15, . . . Find the difference of each
term and the term before
it.
5
The sequence is not arithmetic.
Example: Identifying Arithmetic Sequences
Determine if the sequence could be arithmetic.
If so, give the common difference.
C. 65, 60, 55, 50, 45, . . .
65
–5
60
55
–5
50
–5
45, . . . Find the difference of each
term and the term before
it.
–5
The sequence could be arithmetic with a common
difference of –5.
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, . . .
5.7
5.8
0.1
5.9
0.1
0.1
6
6.1, . . . Find the difference of each
term and the term before
it.
0.1
The sequence could be arithmetic with a common
difference of 0.1.
Example: Identifying Arithmetic Sequences
Determine if the sequence could be arithmetic.
If so, give the common difference.
E. 1, 0, -1, 0, 1, . . .
1
0
–1
–1
–1
0
1
1, . . . Find the difference of each
term and the term before
it.
1
The sequence is not arithmetic.
Try This
Determine if the sequence could be arithmetic.
If so, give the common difference.
A. 1, 2, 3, 4, 5, . . .
1
2
1
3
1
4
1
5, . . . Find the difference of each
term and the term before
it.
1
The sequence could be arithmetic with a common
difference of 1.
Try This
Determine if the sequence could be arithmetic.
If so, give the common difference.
B. 1, 3, 7, 8, 12, …
1
3
2
7
4
8
1
12, . . . Find the difference of each
term and the term before
it.
4
The sequence is not arithmetic.
Try This
Determine if the sequence could be arithmetic.
If so, give the common difference.
C. 11, 22, 33, 44, 55, . . .
11
11
22
11
33
11
55, . . .Find the difference of each
term and the term before
it.
11
44
The sequence could be arithmetic with a
common difference of 11.
Try This
Determine if the sequence could be arithmetic.
If so, give the common difference.
D. 1, 1, 1, 1, 1, 1, . . .
1
1
0
1
0
1
0
1, . . . Find the difference of each
term and the term before
it.
0
The sequence could be arithmetic with a
common difference of 0.
Try This
Determine if the sequence could be arithmetic.
If so, give the common difference.
E. 2, 4, 6, 8, 9, . . .
2
4
2
6
2
8
2
9, . . . Find the difference of each
term and the term before
it.
1
The sequence is not arithmetic.
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
aThe
a1 + (nThe
–NUMBER
1)d.
The term we
FIRST
The COMMON
n =
are looking
for.
term in the
series.
of the term
we are
looking for.
DIFFERENCE.
Example
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
Example
Find the given term in the arithmetic
sequence.
B. 18th term: 100, 93, 86, 79, . . .
an = a1 + (n – 1)d
a18 = 100 + (18 – 1)(–7)
a18 = -19
Example
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
Example
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
Try This
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
Try This
Find the given term in the arithmetic
sequence.
B. 50th term: 100, 93, 86, 79, . . .
an = a1 + (n – 1)d
a50 = 100 + (50 – 1)(-7)
a50 = –243
Try This
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
Try This
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
You can use the formula for the nth term
of an arithmetic sequence to solve for
other variables.
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:
20.5, 21, 21.5, 22, . . .
a1 = 20.5 Let a1 = 20.5 = money after first sale.
d = 0.5
an = 63.5
Example
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
63.5 = 20.5 + (n – 1)(0.5)
Solve for n.
63.5 = 20.5 + 0.5n – 0.5
Distributive Property.
63.5 = 20 + 0.5n
Combine like terms.
43.5 = 0.5n
Subtract 20 from
both sides.
Divide both sides by 0.5.
87 = n
During the bake sale, 87 items are sold in order for
the cash box to contain $63.50.
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 = 10.25 = money after
first sale.
d = 0.25
an = 40
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 = 10.25 + (n – 1)(0.25) Solve for n.
40 = 10.25 + 0.25n – 0.25 Distributive Property.
40 = 10 + 0.25n
Combine like terms.
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.
Lesson Quiz
Determine if each sequence could be arithmetic.
If so, give the common difference.
1. 42, 49, 56, 63, 70, . . . yes; 7
2. 1, 2, 4, 8, 16, 32, . . . no
Find the given term in each arithmetic sequence.
3. 15th term: a1 = 7, d = 5 77
4. 24th term: 1, 5 , 3 , 7 , 2 27 , or 6.75
4
4 2 4
5. 52nd term: a1 = 14.2; d = –1.2 –47