Slide 8- 2 - Edmond Public Schools
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8.1
Sequences
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall
Quick Review
Let f ( x)
x
. Find the values of f .
x4
1. f (5)
2. f (-1)
Evaluate the expression a n 1 d for the given values of
a, n, and d .
3. a -2, n 2, d 3
4. a 1, n 2, d 2
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall
Slide 8- 2
Quick Review
n -1
Evaluate the expression ar for the given values of a, r , and n.
1
5. a , r 2, n 3
2
6. a 2, r 1.5, n 4
Find the value of the limit.
2x 2
7. lim
4x x 1
sin 4 x
8. lim
x
2
x
2
x 0
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall
Slide 8- 3
Quick Review Solutions
x
. Find the values of f .
x4
5
1. f (5)
9
1
2. f (-1) 3
Let f ( x)
Evaluate the expression a n 1 d for the given values of
a, n, and d .
3. a -2, n 2, d 3 1
4. a 1, n 2, d 2 -5
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall
Slide 8- 4
Quick Review Solutions
n -1
Evaluate the expression ar for the given values of a, r , and n.
1
5. a , r 2, n 3 2
2
6. a 2, r 1.5, n 4 -6.75
Find the value of the limit.
2x 2
1
7. lim
4x x 1 2
sin 4 x
8. lim
4
x
2
x
2
x 0
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall
Slide 8- 5
What you’ll learn about
Defining a Sequence
Arithmetic and Geometric Sequences
Graphing a Sequence
Limit of a Sequence
…and why
Sequences arise frequently in mathematics and
applied fields.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall
Slide 8- 6
Defining a Sequence
A sequence a is a list of numbers written in an explicit order.
n
For example: a a , a , a ,..., a ,... , where a is the first term
n
1
2
3
n
1
and a is the nth term of the sequence.
n
Let a , a , a ,..., a ,... be a function with domain the set of positive
1
2
3
n
integers and range a , a , a ,..., a ,.... If the domain is finite, then
1
2
3
n
the sequence is a finite sequence. If the domain is infinite, then
the sequence is an infinite sequence.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall
Slide 8- 7
Example Defining a Sequence Explicitly
Find the first four terms and the 100th term of the sequence a
n
1
where a
.
n 2
Set n equal to 1, 2, 3, 4, and 100.
n
n
2
1 1
1 2 3
1 1
2 2 6
1
11
1
18
1
10, 002
1
a
1
2
2
a
2
a
3
a
a
4
100
2
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall
Slide 8- 8
Example Defining a Sequence Recursively
Find the first three terms and the seventh term for the sequence defined
recursively by the conditions: b 4 and b b 2 for all n 2.
1
n 1
n
Since b 4 and b b 2, you can find b 2, b 0, and b 8.
1
n
n 1
2
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall
3
7
Slide 8- 9
Arithmetic Sequence
A sequence a is an arithmetic sequence if it can be written in the
n
form a, a d , a 2d ,..., a n -1 d ,... for some constant d .
The number d is the common difference.
Each term in an arithmetic sequence can be obtained recursively
from its preceeding term by adding d :
a a d for all n 2.
n
n 1
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall
Slide 8- 10
Example Defining Arithmetic Sequences
Given the arithmetic sequence: 3, 1, 5, 9, ... find
(a) the common difference,
(b) the ninth term,
(c) a recursive rule for the nth term,
(d) an explicit rule for the nth term.
(a) The common difference between terms is 4.
(b) a 3 9 1 (4) 29.
9
(c) a 3 and a a 4 for all n 2.
1
n
n 1
(d) a 3 (n 1)(4)
n
4n 7
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall
Slide 8- 11
Geometric Sequence
A sequence a is a geometric sequence if it can be written in the
n
form a, a r , a r ,..., a r ,... for some nonzero constant r.
2
n 1
The number r is the common ratio.
Each term in a geometric sequence can be obtained recursively
from its preceeding term by multiplying by r:
a a r for all n 2.
n
n 1
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall
Slide 8- 12
Example Defining Geometric Sequences
For the geometric sequence 1, 3, 9, 27, ..., find
(a) the common ratio,
(b) the tenth term,
(c) a recursive rule for the nth term,
(d) an explicit rule for the nth term.
(a) The common ratio is 3.
(b) a (1) (3) 19683
9
10
(c) The sequence is defined recursively by a 1 and a 3 a
1
n
n 1
for n 2.
(d) The sequence is defined explicitly by a 1 3 3 .
n1
n1
n
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall
Slide 8- 13
Example Constructing a Sequence
The second and fifth term of a geometric sequence are -6 and 48,
respectively. Find the first term, common ratio and an explicit rule
for the nth term.
4
ar
48
ar
6
1
1
r 8
r 2
Then a r 6 means that a 3.
3
1
1
The sequence is defined explicitly: a (3) 2 (1)
n 1
n
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall
n 1
3 2
n 1
Slide 8- 14
Example Graphing a Sequence Using
Parametric Mode
Draw a graph of the sequence a with a 1
n
n
n
n 1
, n 1, 2,... .
n
T 1
Let X =T,Y =( 1)
, and graph in dot mode. Set T 1,
T
T 20, and T 1. Choose X 0, X 20, X 2, Y = 2,
T
1T
1T
max
min
step
min
max
scl
min
Y 2, Y 1, and draw the graph.
max
scl
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall
Slide 8- 15
Example Graphing a Sequence Using
Sequence Graphing Mode
Graph the sequence defined recursively by b 4 and
1
b b 2 for all n 2.
n
n -1
Set the graph in sequence graphing mode and dot mode.
Replace b by u (n). Select nMin 1, u (n) u (n 1) 2,
n
and u (nMin) 4.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall
Slide 8- 16
Example Graphing a Sequence Using
Sequence Graphing Mode
Set nMin 1, nMax 10, PlotStart 1, PlotStep 1, and graph in the
0,10 by -5,25 viewing window
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall
Slide 8- 17
Limit
Let L be a real number. The sequence a has limit L as n approaches
n
if, given any positive number , there is a positive number M such that for
all n M we have a - L .
n
We write lim a L and say that the sequence converges to L.
n
n
Sequences that do not have limits diverge.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall
Slide 8- 18
Properties of Limits
If L and M are real numbers and lim a L and lim b M , then
n
n
n
n
1. Sum Rule: lim a b L M
n
n
n
2. Difference Rule: lim a b L M
n
n
n
3. Product Rule: lim a b L M
n
n
n
4. Constant Multiple Rule: lim ca c L
n
n
a L
5. Quotient Rule: lim , M 0
b M
n
n
n
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall
Slide 8- 19
Example Finding the Limit of a Sequence
Determine whether the sequence converges or diverges. If it converges,
find its limit. a
n
2n 1
n
Analytically, using the Properties of Limits:
lim
n
2n 1
1
lim 2
n
n
n
1
lim(2) lim
n
20 2
n
n
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall
Slide 8- 20
The Sandwich Theorem for Sequences
If lim a lim c L and if there is an integer N for which
n
n
n
n
a b c for all n N , then lim b L.
n
n
n
n
n
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall
Slide 8- 21
Absolute Value Theorem
Consider the sequence a . If lim a 0, then lim a 0.
n
n
n
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall
n
n
Slide 8- 22