Ch. 11 – Sequences & Series

11.1 – Sequences as Functions

• Arithmetic sequence -

• 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

• 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, …

• 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

• 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

• 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

• 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

• 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

• 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

• 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

• 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

• 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

• 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

• 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

• 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

• 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

• 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

b) 23, 18, 13, …

b) 23, 18, 13, … 23

b) 23, 18, 13, … 23 - 5

b) 23, 18, 13, … 23 18 - 5

b) 23, 18, 13, … 23 18 - 5 - 5

b) 23, 18, 13, … 23 18 13 - 5 - 5

b) 23, 18, 13, … 23 18 13 - 5 - 5 - 5

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

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

• Geometric sequence

• 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

• 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, ___

• 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

• 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

• 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

• 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

• 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

• 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

• 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,

• 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

• 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

• 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

• 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

• 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

Ex. 3 Determine whether each sequence is arithmetic, geometric, or neither. Then graph each sequence.

a) 16, 24, 36, 54 …

Ex. 3 Determine whether each sequence is arithmetic, geometric, or neither. Then graph each sequence.

a) 16, 24, 36, 54 … 24-16=8

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

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

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

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

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

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

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.

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

b) 1, 4, 9, 16 …

c) 23, 17, 11, 5 …

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