Ch. 11 – Sequences & Series
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Transcript of Ch. 11 – Sequences & Series
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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
![Page 4: Ch. 11 – Sequences & Series](https://reader035.fdocuments.us/reader035/viewer/2022081513/568132aa550346895d99521a/html5/thumbnails/4.jpg)
• 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, …
![Page 5: Ch. 11 – Sequences & Series](https://reader035.fdocuments.us/reader035/viewer/2022081513/568132aa550346895d99521a/html5/thumbnails/5.jpg)
• 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
![Page 6: Ch. 11 – Sequences & Series](https://reader035.fdocuments.us/reader035/viewer/2022081513/568132aa550346895d99521a/html5/thumbnails/6.jpg)
• 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
![Page 7: Ch. 11 – Sequences & Series](https://reader035.fdocuments.us/reader035/viewer/2022081513/568132aa550346895d99521a/html5/thumbnails/7.jpg)
• 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
![Page 8: Ch. 11 – Sequences & Series](https://reader035.fdocuments.us/reader035/viewer/2022081513/568132aa550346895d99521a/html5/thumbnails/8.jpg)
• 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
![Page 9: Ch. 11 – Sequences & Series](https://reader035.fdocuments.us/reader035/viewer/2022081513/568132aa550346895d99521a/html5/thumbnails/9.jpg)
• 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
![Page 10: Ch. 11 – Sequences & Series](https://reader035.fdocuments.us/reader035/viewer/2022081513/568132aa550346895d99521a/html5/thumbnails/10.jpg)
• 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
![Page 11: Ch. 11 – Sequences & Series](https://reader035.fdocuments.us/reader035/viewer/2022081513/568132aa550346895d99521a/html5/thumbnails/11.jpg)
• 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
![Page 12: Ch. 11 – Sequences & Series](https://reader035.fdocuments.us/reader035/viewer/2022081513/568132aa550346895d99521a/html5/thumbnails/12.jpg)
• 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
![Page 13: Ch. 11 – Sequences & Series](https://reader035.fdocuments.us/reader035/viewer/2022081513/568132aa550346895d99521a/html5/thumbnails/13.jpg)
• 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
![Page 14: Ch. 11 – Sequences & Series](https://reader035.fdocuments.us/reader035/viewer/2022081513/568132aa550346895d99521a/html5/thumbnails/14.jpg)
• 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
![Page 15: Ch. 11 – Sequences & Series](https://reader035.fdocuments.us/reader035/viewer/2022081513/568132aa550346895d99521a/html5/thumbnails/15.jpg)
• 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
![Page 16: Ch. 11 – Sequences & Series](https://reader035.fdocuments.us/reader035/viewer/2022081513/568132aa550346895d99521a/html5/thumbnails/16.jpg)
• 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
![Page 17: Ch. 11 – Sequences & Series](https://reader035.fdocuments.us/reader035/viewer/2022081513/568132aa550346895d99521a/html5/thumbnails/17.jpg)
• 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
![Page 18: Ch. 11 – Sequences & Series](https://reader035.fdocuments.us/reader035/viewer/2022081513/568132aa550346895d99521a/html5/thumbnails/18.jpg)
• 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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b) 23, 18, 13, … 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
![Page 30: Ch. 11 – Sequences & Series](https://reader035.fdocuments.us/reader035/viewer/2022081513/568132aa550346895d99521a/html5/thumbnails/30.jpg)
• 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
![Page 42: Ch. 11 – Sequences & Series](https://reader035.fdocuments.us/reader035/viewer/2022081513/568132aa550346895d99521a/html5/thumbnails/42.jpg)
• 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=12 NOT 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=12 NOT 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=12 NOT 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=12 NOT ARITHMETIC
24 = 1.5, 36 = 1.5 GEOMETRIC
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=12 NOT ARITHMETIC
24 = 1.5, 36 = 1.5 GEOMETRIC
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=12 NOT ARITHMETIC
24 = 1.5, 36 = 1.5 GEOMETRIC
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=12 NOT ARITHMETIC
24 = 1.5, 36 = 1.5 GEOMETRIC
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=12 NOT ARITHMETIC 4 = 4 , 9 = 2.25 NOT GEOMETRIC
1 4SO NEITHER
c) 23, 17, 11, 5 … 17-23=-6, 11-17=-6 ARITHMETIC