1. Sequences

2. Objectives: Essential Question: Sequences
Identify, analyze, and create linear relations, sequences, and functions using symbols, graphs, tables, diagrams, and written descriptions.
Essential Question:
What is the difference between arithmetic and geometric sequences?

3. Vocabulary: Sequences
Sequence: a list or set of numbers in a certain order.
Arithmetic Sequence: a numerical pattern that increases or decreases at a constant rate or value. The difference between successive terms of the sequence is constant. (example: 2, 5, 8, 11, 14…)
Geometric Sequence: a sequence in which each term can be found by multiplying the previous term by the same number.

4. 1) Arithmetic Sequences 2) Geometric Sequences
What Are They:
We use sequences all the time in our day to day, but in math class we will study two specific types of sequences:
1) Arithmetic Sequences
2) Geometric Sequences

5. 4, 8, 12, 16, … Arithmetic Sequences: Sequences
In an arithmetic sequence, each term is found by adding the same number to the previous term.
Example:
4, 8, 12, 16, …
+ 4
+ 4
+ 4

6. 2, 4, 8, 16, … Geometric Sequences: Sequences
In an geometric sequence, each term is found by multiplying the previous term by the same number.
Example:
2, 4, 8, 16, …
x 2
x 2
x 2

7. 0, 9, 18, 27, … Example 1: Identifying Sequences Sequences
Describe the pattern in the sequence and identify it as arithmetic, geometric, or neither.
Example 1: 0, 9, 18, 27, …
0, 9, 18, 27, … +9 +9 +9
Because each term is found by adding 9 to the previous term, this is an arithmetic sequence.

8. 2, 6, 18, 54, … Example 2: Identifying Sequences Sequences
Describe the pattern in the sequence and identify it as arithmetic, geometric, or neither.
Example 2: 2, 6, 18, 54, …
2, 6, 18, 54, …
x 3
x 3
x 3
Because each term is found by multiplying the previous number by 3, this is an geometric sequence.

9. 0, 2, 6, 12, … Example 3: Identifying Sequences Sequences
Describe the pattern in the sequence and identify it as arithmetic, geometric, or neither.
Example 3: 0, 2, 6, 12, …
0, 2, 6, 12, … +2 +4 +6
Because each term is found by adding two more than what was added to the previous term, this is neither.

10. 5, 5.4, 5.8, 6.2, __ , __ , __ Example 4: Identifying Sequences
Identify as arithmetic, geometric, or neither and then write the next three terms.
Example 4: 5, 5.4, 5.8, 6.2, …
5, 5.4, 5.8, 6.2, __ , __ , __
+0.4
+0.4
+0.4
+0.4
+0.4
+0.4
This is an arithmetic sequence, in which 0.4 is added to each term. The next terms are 6.6, 7.0, and 7.4.

11. Real World: Construction Work
Sequences
Real World: Construction Work
The table below shows typical costs for a construction company to rent a crane for one, two, three, or four months. Assuming that the arithmetic sequence continues, how much would it cost to rent the crane for 24 months?
Months
Cost($) 1
75,000 2
90,000 3
105,000 4
120,000

12. Real World: Construction Work
Sequences
Real World: Construction Work
Since the difference between any two successive costs is $15,000, the costs form an arithmetic sequence with common difference 15,000. Our solution should include an initial cost of $75,000 and an increase of $15,000 per month.
Solution 1:
Since the difference between any two successive costs is $15,000, the costs form an arithmetic sequence with common difference 15,000.
Months
Cost($) 1
75,000 2
90,000 3
105,000 4
120,000
We had an initial $75,000 + $15,000 for 23 months = $240,000

13. Real World: Construction Work
Sequences
Real World: Construction Work
However, algebraically we would solve this problem using the following:

14. Independent Practice:
Sequences
Independent Practice:
Determine what kind of sequence and the rule.
1. 1, 3, 5, 7, …
2. 5, 15, 45, 135, …
3. 0.5, 1.5, 4.5, 13.5, …
4. 11, 22, 33, 44, …
5. 1, 2, 6, 24, …
Arithmetic (+ 2)
Geometric (x 3)
Geometric (x 3)
Arithmetic (+ 11)
Neither

15. Summary: Sequences Arithmetic Sequences Geometric Sequences
We have studied two kinds of sequences:
Arithmetic Sequences
Each term is found by adding the same number to the previous term:
5, 10, 15, 20, 25, … (+ 5)
Geometric Sequences
Each term is found by multiplying the previous term by the same number:
3, 12, 48, 182, 728, … (x 4)

16. Sequences
HOMEWORK