course 3 3-6 arithmetic sequences 3-6 arithmetic sequences course 3 problem of the day problem of...
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Course 3
3-6 Arithmetic Sequences
3-6 Arithmetic Sequences
Course 3
Problem of the Day
Lesson Presentation
Warm Up
Course 3
3-6 Arithmetic Sequences
Warm UpUse the table to write an equation.
1.
2.
3.
y = 5xx 1 2 3 4
y 5 10 15 20
x 1 2 3 4
y –2.5 –5 –7.5 –10
x 1 2 3 4
y 5 8 11 14
y = –2.5x
y = 3x + 2
Course 3
3-6 Arithmetic Sequences
Problem of the Day
A movie ticket at a certain theater costs $4.50 for a child and $8.75 for an adult. How much will it cost for a family of two adults and three children to see a movie?$31
Course 3
3-6 Arithmetic Sequences
Learn to identify and evaluate arithmetic sequences.
Course 3
3-6 Arithmetic Sequences
Vocabulary
sequence
term
arithmetic sequence
common difference
Course 3
3-6 Arithmetic Sequences
A sequence is an ordered list of numbers or objects, called terms. 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.
Course 3
3-6 Arithmetic Sequences
The terms decrease by 2.
Additional Example 1: Finding the Common Difference in an Arithmetic Sequence
Find the common difference in each arithmetic sequence.A. 11, 9, 7, 5, …
11, 9, 7, 5, …
–2 –2 –2The common difference is 2.
The terms increase by 1.25.
B. 2.5, 3.75, 5, 6.25, …
2.5, 3.75, 5, 6.25, …
+1.25 +1.25 +1.25
The common difference is 1.25.
Course 3
3-6 Arithmetic Sequences
The terms increase by 4.
Check It Out: Example 1
Find the common difference in each arithmetic sequence.A. 2, 6, 10, 14, …
2, 6, 10, 14, …
+4 +4 +4
The common difference is 4.
The terms increase by 2.5.
B. 2.5, 5, 7.5, 10, …
2.5, 5, 7.5, 10, …
+2.5 +2.5 +2.5
The common difference is 2.5.
Course 3
3-6 Arithmetic Sequences
Additional Example 2: Finding Missing Terms in an Arithmetic Sequence
Each term is 5 more than the previous term.
Use the common difference to find the next three terms.
Find the next three terms in the arithmetic sequence –8, –3, 2, 7, ...
7 + 5 = 12
12 + 5 = 17
17 + 5 = 22
The next three terms are 12, 17, and 22.
Course 3
3-6 Arithmetic Sequences
Check It Out: Example 2
Each term is 3 more than the previous term.
Use the common difference to find the next three terms.
Find the next three terms in the arithmetic sequence –9, –6, –3, 0, ...
0 + 3 = 3
3 + 3 = 6
6 + 3 = 9
The next three terms are 3, 6, and 9.
Course 3
3-6 Arithmetic Sequences
Additional Example 3A: Identifying Functions in Arithmetic Sequences
Multiply n by 6.
Find a function that describes each arithmetic sequence.
6, 12, 18, 24, …
n n • 6 y
1
2
3
4
n
y = 6n
1 • 6 6
2 • 6 12
3 • 6 18
4 • 6 24
n • 6 6n
Course 3
3-6 Arithmetic Sequences
Additional Example 3B: Identifying Functions in Arithmetic Sequences
Multiply n by 4.
Find a function that describes each arithmetic sequence.
–4, –8, –12, –16, …
n n • (– 4) y
1
2
3
4
n
y = –4n
1 • (–4) –4
2 • (–4) –8
3 • (–4) –12
4 • (–4) –16
n • (–4) –4n
Course 3
3-6 Arithmetic Sequences
Check It Out: Example 3A
Multiply n by 3.
Find a function that describes each arithmetic sequence.
3, 6, 9, 12, …
n n • 3 y
1
2
3
4
n
y = 3n
1 • 3 3
2 • 3 6
3 • 3 9
4 • 3 12
n • 3 3n
Course 3
3-6 Arithmetic Sequences
Check It Out: Example 3B
Multiply n by 7.
Find a function that describes each arithmetic sequence.
–7, –14, –21, –28, …
n n • (7) y
1
2
3
4
n
y = 7n
1 • (7) 72 • (7) 14
3 • (7) 21
4 • (7) 28
n • (7) 7n
Course 3
3-6 Arithmetic Sequences
A DVD costs $3.95 to rent, plus $0.45 for each day it is returned late. Find a function that describes the arithmetic sequence, and then find the total cost of renting the DVD and returning it 9 days late.
Additional Example 4: Application
Multiply n by $0.45, and then add the $3.95 rental fee.
n 3.95 + 0.45n y
1
2
3
4
n
3.95 + 0.45(1) 4.40
3.95 + 0.45(2) 4.853.95 + 0.45(3) 5.303.95 + 0.45(4) 5.753.95 + 0.45(5) 0.45n + 3.95
Course 3
3-6 Arithmetic Sequences
A DVD costs $3.95 to rent, plus $0.45 for each day it is returned late. Find a function that describes the arithmetic sequence, and then find the total cost of renting the DVD and returning it 9 days late.
Additional Example 4 Continued
Write a function to find the 9th term.0.45n + 3.95
Substitute 9 for n.0.45(9) + 3.95
4.05 + 3.95 Multiply.
8.00 Add.
It will cost $8.00 to rent a movie and return it 9 days late.
Course 3
3-6 Arithmetic Sequences
A personal pizza with cheese costs $3.99 to buy, plus $0.25 for each additional topping. Find a function that describes the arithmetic sequence, and then find the total cost of buying a personal pizza with 7 additional toppings.
Check It Out: Example 4
Multiply n by $0.25, and then add the $3.99 pizza cost.
n 3.99 + 0.25n y
1
2
3
4
n
3.99 + 0.25(1) 4.24
3.99 + 0.25(2) 4.493.99 + 0.25(3) 4.743.99 + 0.25(4) 4.993.99 + 0.25(5) 0.25n + 3.99
Course 3
3-6 Arithmetic Sequences
Check It Out: Example 4 Continued
Write a function to find the 7th term.0.25n + 3.99
Substitute 7 for n.0.25(7) + 3.99
1.75 + 3.99 Multiply.
5.74 Add.
It will cost $5.74 to buy a personal pizza with 7 additional toppings.
A personal pizza with cheese costs $3.99 to buy, plus $0.25 for each additional topping. Find a function that describes the arithmetic sequence, and then find the total cost of buying a personal pizza with 7 additional toppings.
Course 3
3-6 Arithmetic Sequences
Lesson Quiz: Part 1
Find the common difference in each arithmetic sequence.
1. 4, 2, 0, –2, …
2. , 2, , , …
Find the next three terms in each arithmetic sequence.
3. 18, 13, 8, 3, …
4. 3.6, 5, 6.4, 7.8, …
–2
2 34 3 8 3 10 3
–2, –7, –12
9.2, 10.6, 12
Course 3
3-6 Arithmetic Sequences
Find a function that describes the arithmetic sequence.
5. –5, –10, –15, –20, … Possible Answer: y = –5n
6. –1, 2, 5, 8, … Possible Answer: y = 3n – 4
Lesson Quiz: Part 2
7. A runner finishes a lap in 55 seconds. Her goal is to decrease her time by two seconds every week. Find a function that describes the arithmetic sequence and find how many seconds her lap should be after 12 weeks of training.
Possible Answer: s = -2w + 55; 31 seconds