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Ch. 11 – Sequences & Series 11.1 – Sequences as Functions

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Arithmetic sequence -

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Arithmetic sequence – a sequence of numbers in which each term after the first is found by adding or subtracting a constant number, called the “common difference” d

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Ex. 1 Find the next four terms. a) 36, 42, 48, …

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Arithmetic sequence – a sequence of numbers in which each term after the first is found by adding or subtracting a constant number, called the “common difference” d Ex. 1 Find the next four terms. a) 36, 42, 48, … 36

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Arithmetic sequence – a sequence of numbers in which each term after the first is found by adding or subtracting a constant number, called the “common difference” d Ex. 1 Find the next four terms. a) 36, 42, 48, … 36 + 6

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Arithmetic sequence – a sequence of numbers in which each term after the first is found by adding or subtracting a constant number, called the “common difference” d Ex. 1 Find the next four terms. a) 36, 42, 48, … 36 42 + 6

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Arithmetic sequence – a sequence of numbers in which each term after the first is found by adding or subtracting a constant number, called the “common difference” d Ex. 1 Find the next four terms. a) 36, 42, 48, … 36 42 + 6 + 6

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Arithmetic sequence – a sequence of numbers in which each term after the first is found by adding or subtracting a constant number, called the “common difference” d Ex. 1 Find the next four terms. a) 36, 42, 48, … 36 42 48 + 6 + 6

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Arithmetic sequence – a sequence of numbers in which each term after the first is found by adding or subtracting a constant number, called the “common difference” d Ex. 1 Find the next four terms. a) 36, 42, 48, … 36 42 48 + 6 + 6 + 6

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Arithmetic sequence – a sequence of numbers in which each term after the first is found by adding or subtracting a constant number, called the “common difference” d Ex. 1 Find the next four terms. a) 36, 42, 48, … 36 42 48 54 + 6 + 6 + 6

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Arithmetic sequence – a sequence of numbers in which each term after the first is found by adding or subtracting a constant number, called the “common difference” d Ex. 1 Find the next four terms. a) 36, 42, 48, … 36 42 48 54 + 6 + 6 + 6 + 6

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Arithmetic sequence – a sequence of numbers in which each term after the first is found by adding or subtracting a constant number, called the “common difference” d Ex. 1 Find the next four terms. a) 36, 42, 48, … 36 42 48 54 60 + 6 + 6 + 6 + 6

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Arithmetic sequence – a sequence of numbers in which each term after the first is found by adding or subtracting a constant number, called the “common difference” d Ex. 1 Find the next four terms. a) 36, 42, 48, … 36 42 48 54 60 + 6 + 6 + 6 + 6 + 6

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Arithmetic sequence – a sequence of numbers in which each term after the first is found by adding or subtracting a constant number, called the “common difference” d Ex. 1 Find the next four terms. a) 36, 42, 48, … 36 42 48 54 60 66 + 6 + 6 + 6 + 6 + 6

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Arithmetic sequence – a sequence of numbers in which each term after the first is found by adding or subtracting a constant number, called the “common difference” d Ex. 1 Find the next four terms. a) 36, 42, 48, … 36 42 48 54 60 66 + 6 + 6 + 6 + 6 + 6 + 6

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Arithmetic sequence – a sequence of numbers in which each term after the first is found by adding or subtracting a constant number, called the “common difference” d Ex. 1 Find the next four terms. a) 36, 42, 48, … 36 42 48 54 60 66 72 + 6 + 6 + 6 + 6 + 6 + 6

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Arithmetic sequence – a sequence of numbers in which each term after the first is found by adding or subtracting a constant number, called the “common difference” d Ex. 1 Find the next four terms. a) 36, 42, 48, … 36 42 48 54 60 66 72 + 6 + 6 + 6 + 6 + 6 + 6

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b) 23, 18, 13, …

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23

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b) 23, 18, 13, … 23 - 5

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b) 23, 18, 13, … 23 18 - 5

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b) 23, 18, 13, … 23 18 - 5 - 5

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b) 23, 18, 13, … 23 18 13 - 5 - 5

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b) 23, 18, 13, … 23 18 13 - 5 - 5 - 5

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b) 23, 18, 13, … 23 18 13 8 - 5 - 5 - 5

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b) 23, 18, 13, … 23 18 13 8 3 -2 -7 - 5 - 5 - 5 - 5 - 5 - 5

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Geometric sequence

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Geometric sequence – a sequence of numbers in which each term after the first is found by multiplying the previous terms by a constant number, called the “common ratio” r

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Geometric sequence – a sequence of numbers in which each term after the first is found by multiplying the previous terms by a constant number, called the “common ratio” r Ex. 2 Find the next term. a) 8, 24, 72, ___

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Geometric sequence – a sequence of numbers in which each term after the first is found by multiplying the previous terms by a constant number, called the “common ratio” r Ex. 2 Find the next term. a) 8, 24, 72, ___ 24 8

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Geometric sequence – a sequence of numbers in which each term after the first is found by multiplying the previous terms by a constant number, called the “common ratio” r Ex. 2 Find the next term. a) 8, 24, 72, ___ ·3

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Geometric sequence – a sequence of numbers in which each term after the first is found by multiplying the previous terms by a constant number, called the “common ratio” r Ex. 2 Find the next term. a) 8, 24, 72, ___ ·3 72 24

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Geometric sequence – a sequence of numbers in which each term after the first is found by multiplying the previous terms by a constant number, called the “common ratio” r Ex. 2 Find the next term. a) 8, 24, 72, ___ ·3 ·3

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Geometric sequence – a sequence of numbers in which each term after the first is found by multiplying the previous terms by a constant number, called the “common ratio” r Ex. 2 Find the next term. a) 8, 24, 72, 72·3 ·3 ·3

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Geometric sequence – a sequence of numbers in which each term after the first is found by multiplying the previous terms by a constant number, called the “common ratio” r Ex. 2 Find the next term. a) 8, 24, 72, 216 ·3 ·3

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Geometric sequence – a sequence of numbers in which each term after the first is found by multiplying the previous terms by a constant number, called the “common ratio” r Ex. 2 Find the next term. a) 8, 24, 72, 216 ·3 ·3 b) 270, 90, 30,

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Geometric sequence – a sequence of numbers in which each term after the first is found by multiplying the previous terms by a constant number, called the “common ratio” r Ex. 2 Find the next term. a) 8, 24, 72, 216 ·3 ·3 b) 270, 90, 30, ÷3

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Geometric sequence – a sequence of numbers in which each term after the first is found by multiplying the previous terms by a constant number, called the “common ratio” r Ex. 2 Find the next term. a) 8, 24, 72, 216 ·3 ·3 b) 270, 90, 30, ·⅓

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Geometric sequence – a sequence of numbers in which each term after the first is found by multiplying the previous terms by a constant number, called the “common ratio” r Ex. 2 Find the next term. a) 8, 24, 72, 216 ·3 ·3 b) 270, 90, 30, ·⅓ ·⅓

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Geometric sequence – a sequence of numbers in which each term after the first is found by multiplying the previous terms by a constant number, called the “common ratio” r Ex. 2 Find the next term. a) 8, 24, 72, 216 ·3 ·3 b) 270, 90, 30, 30·⅓ ·⅓ ·⅓

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Geometric sequence – a sequence of numbers in which each term after the first is found by multiplying the previous terms by a constant number, called the “common ratio” r Ex. 2 Find the next term. a) 8, 24, 72, 216 ·3 ·3 b) 270, 90, 30, 10 ·⅓ ·⅓

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Ex. 3 Determine whether each sequence is arithmetic, geometric, or neither. Then graph each sequence. a) 16, 24, 36, 54 …

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Ex. 3 Determine whether each sequence is arithmetic, geometric, or neither. Then graph each sequence. a) 16, 24, 36, 54 … 24-16=8

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Ex. 3 Determine whether each sequence is arithmetic, geometric, or neither. Then graph each sequence. a) 16, 24, 36, 54 … 24-16=8, 36-24=12

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Ex. 3 Determine whether each sequence is arithmetic, geometric, or neither. Then graph each sequence. a) 16, 24, 36, 54 … 24-16=8, 36-24=12NOT ARITHMETIC

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Ex. 3 Determine whether each sequence is arithmetic, geometric, or neither. Then graph each sequence. a) 16, 24, 36, 54 … 24-16=8, 36-24=12NOT ARITHMETIC 24 = 1.5 16

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Ex. 3 Determine whether each sequence is arithmetic, geometric, or neither. Then graph each sequence. a) 16, 24, 36, 54 … 24-16=8, 36-24=12NOT ARITHMETIC 24 = 1.5, 36 = 1.5 16 24

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Ex. 3 Determine whether each sequence is arithmetic, geometric, or neither. Then graph each sequence. a) 16, 24, 36, 54 … 24-16=8, 36-24=12NOT ARITHMETIC 24 = 1.5, 36 = 1.5GEOMETRIC 16 24

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Ex. 3 Determine whether each sequence is arithmetic, geometric, or neither. Then graph each sequence. a) 16, 24, 36, 54 … 24-16=8, 36-24=12NOT ARITHMETIC 24 = 1.5, 36 = 1.5GEOMETRIC 16 24

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Ex. 3 Determine whether each sequence is arithmetic, geometric, or neither. Then graph each sequence. a) 16, 24, 36, 54 … 24-16=8, 36-24=12NOT ARITHMETIC 24 = 1.5, 36 = 1.5GEOMETRIC 16 24 ***NOTE: You do not need to graph. However, open your book to p.684 and look at top of page.

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Ex. 3 Determine whether each sequence is arithmetic, geometric, or neither. Then graph each sequence. a) 16, 24, 36, 54 … 24-16=8, 36-24=12NOT ARITHMETIC 24 = 1.5, 36 = 1.5GEOMETRIC 16 24 ***NOTE: You do not need to graph. However, open your book to p.684 and look at top of page. Arithmetic Sequences = Linear Geometric Sequences = Exponential

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b) 1, 4, 9, 16 … c) 23, 17, 11, 5 …

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b) 1, 4, 9, 16 … 4-1=3, 9-4=12NOT ARITHMETIC 4 = 4, 9 = 2.25NOT GEOMETRIC 1 4 SO NEITHER c) 23, 17, 11, 5 … 17-23=-6, 11-17=-6ARITHMETIC

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