Sequences

Objectives: Essential Question: Sequences 7.5.01 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?

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.

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

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

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

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.

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.

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.

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.

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

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

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

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

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)

Sequences HOMEWORK