chapter 14 14.1 sequences and series definition sequence a sequence is an ordered set of numbers...
TRANSCRIPT
Chapter 14
14.1 Sequences and Series
Definition Sequence
A Sequence is an ordered set of numbers
Definition Infinite Sequence
An Infinite Sequence is an ordered set of numbers that continues on forever
Ex: 3, 6, 7, 9, …
Each number in a sequence is called a term
Definition Nth term
The Nth term of a sequence is a rule that describes all the terms of a sequence and is sometimes referred to as the General Term of the sequence
Definition Recursive Formula
The Recursive definition of a sequence is a rule that describes the next term in the sequence based on the previous terms
The numbers in sequences are called terms.
You can think of a sequence as a function whose domainis a set of consecutive integers. If a domain is notspecified, it is understood that the domain starts with 1.
The domain gives the relative position of each term.
1 2 3 4 5 DOMAIN:
3 6 9 12 15RANGE:The range gives the terms of the sequence.
This is a finite sequence having the rule
an = 3n,where an represents the nth term of the sequence.
n
an
Writing Terms of Sequences
Write the first six terms of the sequence an = 2n + 3.
SOLUTION
a 1 = 2(1) + 3 = 5 1st term
2nd term
3rd term
4th term
6th term
a 2 = 2(2) + 3 = 7
a 3 = 2(3) + 3 = 9
a 4 = 2(4) + 3 = 11
a 5 = 2(5) + 3 = 13
a 6 = 2(6) + 3 = 15
5th term
Writing Terms of Sequences
Write the first six terms of the sequence f (n) = (–2)
n – 1 .
SOLUTION
f (1) = (–2) 1 – 1 = 1 1st term
2nd term
3rd term
4th term
6th term
f (2) = (–2) 2 – 1 = –2
f (3) = (–2) 3 – 1 = 4
f (4) = (–2) 4 – 1 = – 8
f (5) = (–2) 5 – 1 = 16
f (6) = (–2) 6 – 1 = – 32
5th term
Writing Rules for Sequences
If the terms of a sequence have a recognizable pattern, then you may be able to write a rule for the n th term of the sequence.
Describe the pattern, write the next term, and write a rule for the n th term of the sequence
____ – , , – , , ….13
19
127
181
_ _
1 3
, 1 9
, 1 27
, 1 81
Writing Rules for Sequences
SOLUTION
1 2 3 4n
terms 1243
5
13
4
13
1, 1
3
2, 1
3
3, 1
3
5rewriteterms
13
A rule for the nth term is an =
n
2 6 12 20
Writing Rules for Sequences
SOLUTION
A rule for the nth term is f (n) = n (n+1).
terms
5(5 +1)
Describe the pattern, write the next term, and write a rule for the n th term of the sequence.
2, 6, 12 , 20,….
5
30
1 2 3 4
rewriteterms 1(1 +1) 2(2 +1) 3(3 +1) 4(4 +1)
n
Graphing a Sequence
You can graph a sequence by letting the horizontal axisrepresent the position numbers (the domain) and the vertical axis represent the terms (the range).
Graphing a Sequence
You work in the producedepartment of a grocery storeand are stacking oranges in the shape of square pyramid with ten layers.
• Write a rule for the number of oranges in each layer.
• Graph the sequence.
Graphing a Sequence
SOLUTION
The diagram below shows the first three layers of the
stack. Let an represent the number of oranges in layer n.
n 1 2 3
an 1 = 1 2 4 = 2 2 9 = 3
2
From the diagram, you can see that an = n
2
Graphing a Sequence
Plot the points (1, 1), (2, 4), (3, 9), . . . , (10, 100).
an = n2
. . .
3 + 6 + 9 + 12 + 15 = 3i5
i = 1
FINITE SEQUENCE
FINITE SERIES
3, 6, 9, 12, 15
3 + 6 + 9 + 12 + 15
INFINITE SEQUENCE
INFINITE SERIES
3, 6, 9, 12, 15, . . .
3 + 6 + 9 + 12 + 15 + . . .
You can use summation notation to write a series. Forexample, for the finite series shown above, you can write
When the terms of a sequence are added, the resultingexpression is a series. A series can be finite or infinite.
3 + 6 + 9 + 12 + 15 = 3i5
i = 1
5
i = 1 3i
Is read as “the sum from i equals 1 to 5 of 3i.”
index of summation lower limit of summation
upper limit of summation
USING SERIES
Summation notation is also called sigma notation because it uses the uppercase Greek letter sigma, written .
Summation notation for an infinite series is similarto that for a finite series. For example, for the infiniteseries shown earlier, you can write:
3 + 6 + 9 + 12 + 15 + = 3i
i = 1. . .
The infinity symbol, , indicates that the series continues without end.
The index of summation does not have to be i. Any letter can be used. Also, the index does not have to begin at 1.
Writing Series with Summation Notation
Write the series with summation notation.
5 + 10 + 15 + + 100. . .
SOLUTION
Notice that the first term is 5 (1), the second is 5 (2),the third is 5 (3), and the last is 5 (20). So the termsof the series can be written as:
ai = 5i where i = 1, 2, 3, . . . , 20
The summation notation is 5i.20
i = 1
Writing Series with Summation Notation
Notice that for each term the denominator of the fractionis 1 more than the numerator. So, the terms of the seriescan be written as:
ai = where i = 1, 2, 3, 4 . . . ii + 1
Write the series with summation notation.
SOLUTION
. . .1 2 3 42 3 4 5
+ + + +
The summation notation for the series is
i = 1
ii + 1
.
Writing Series with Summation Notation
The sum of the terms of a finite sequence can be foundby simply adding the terms. For sequences with manyterms, however, adding the terms can be tedious. Formulas for finding the sum of the terms of three special types of sequences are shown next.
Writing Series with Summation Notation
FORMULAS FOR SPECIAL SERIESCONCEPT
SUMMARY
n
i = 1 1 = n
i = n (n + 1)
2
n
i = 1
1
2
3
gives the sum of positive integers from 1 to n .
gives the sum of squares of positive integers from 1 to n.
i 2 = n (n + 1)(2 n + 1)
6
n
i = 1
gives the sum of n 1’s .
Using a Formula for a Sum
RETAIL DISPLAYS How many oranges are in a square pyramid 10 layers high?
Using a Formula for a Sum
SOLUTION
You know from the earlier example that the i th term of the series is given by ai = i
2, where i = 1, 2, 3, . . . , 10.
10
i = 1i
2 = 12+ 22 + + 102 . . .
10(11)(21)=
6
= 385
There are 385 oranges in the stack.
=6
10(10 + 1)(2 • 10 + 1)
HW #14.1aPg 615-616 1-20
Find the nth term of the given sequence
Chapter 14
14.1 Sequences and Series
Write down the first five terms of the following recursively defined sequence. Find an expression for the nth term of the sequence
1 12, 3n na a a
a1 = 2
a2 = 3 + 2 = 5
a3 = 3 + 5 = 8
a4 = 3 + 8 = 11
a5 = 3 + 11 = 14
2 3( 1)na n 3 1n
Write down the first five terms of the following recursively defined sequence. Find an expression for the nth term of the sequence
Definition Partial Sum
The Partial Sum (Sn) of an infinite series is the sum of the first n terms where n is some natural number
For the series given find the first 6 partial sums and determine an expression to find the nth partial sum
1 11na
n n
11
1nSn
In the diagram, suppose each small square has sides 1 unit long, and suppose that the total height and the total width of the entire figure are each n units.
A. Use the area formula for a triangle to find the area of the entire figure (as the sum of the areas of 1 large triangle and n small triangles), in terms of n.
In the diagram, suppose each small square has sides 1 unit long, and suppose that the total height and the total width of the entire figure are each n units.
B. Explain how your answer to part A proves the formula for 1 + 2 + 3 + 4 + … + n
HW #14.1bPg 616 21-44
Chapter 14
14-2 Arithmetic Sequences
Definition Arithmetic Sequence
A sequence in which a constant d can be added to each term to get the next term is called an arithmetic sequence. The constant d is called the common difference.
To decide whether a sequence is arithmetic, find the differences of consecutive terms. an – an-1
To decide whether a sequence is arithmetic, find the differences of consecutive terms. an – an-1
4 4 4 4 3 5 7 9
Decide whether the sequence is arithmetic.
If a sequence is arithmetic we can
1. Simplify the series
2. Create formulas for the nth term
3. Find the sum of the first n terms
Common Difference
Recursive Substituting to get to an
3
10 7
17 7
24 7
…
7
7
7
2 1 7a a 1a
3 2 7a a
1a
2a
3a2 3 7 10a 3 10 7 17a
4a4 3 7a a 4 17 7 24a
15a
50a
na
The first term of an arithmetic sequence is a1. We add d to get the next term, a1 + d. We add d again to get the next term,
(a1 + d) + d, and so on. There is a pattern.
a1 = a1
a2 = a1 + d
a3 = (a1 + d) + d = a1 + 2d
a4 = (a1 + 2d) + d = a1 + 3d
…
an = a1 + (n - l)d
The sequence is arithmetic, a1 = 50 and common difference d = -6.
The sequence is arithmetic, a1 = 2, d = 9
Write a rule for the nth term of the sequence and find a15
Find for the sequence 40a 10, 6, 2,2...
For the sequence what term is 2? 14 13
5, , , ...3 3
An arithmetic sequence has and , what is ?
6 5.5a 11 9a 20a
4 2 8 14 ... 68Find the Sum
Proof Theorem 14-2Let bean arithmeticsequencenA 1 ( 1)nA A d n
1 2 3 ...n nS A A A A
1 2 1...n n n nS A A A A
1 2 1 3 2 12 ( ) ( ) ( ) ... ( )n n n n nS A A A A A A A A
Note: 2 1A A d and 1n nA A d
2 1 1 1( ) ( )n n nA A A d A d A A
All the pairs have a sum 1 nA A
12 ( )n nS n A A Therefore:
1( )2n n
nS A A
Proof Theorem 14-3
Continuing from 14-2:
1( )2n n
nS A A
1 ( 1)nA A d n
1 1( ( ( 1))2n
nS A A d n
1(2 ( 1))2n
nS A d n
Alternate Proof Theorem 14-3Let bean arithmeticsequencenA 1 ( 1)nA A d n
1 2 3 ...n nS A A A A
1 1 1 1( ) ( 2 ) ... ( ( 1))A A d A d A d n
1 (1 2 3 ...( 1))nA d n
1
( 1)
2
n nnA d
12 ( 1)
2
nA d n n
12 ( 1)
2
nA d n n
1(2 ( 1))
2
n A d n
1(2 ( 1))2
nA d n
This is the sum of the first n-1 integers. The sum of the first
n integers =( 1)
2
n n
HW #14.2Pg 622-623 1-35 Odd, 36-41
Chapter 14
14-3 Geometric Sequences
Definition Geometric Sequence
A sequence in which a constant r can be multiplied by each term to get the next term is called a geometric sequence.
The constant r is called the common ratio.
To decide whether a sequence is geometric, find the ratios of consecutive terms.
Decide whether each sequence is geometric.
Decide whether the sequence is arithmetic, geometric, or neither. Explain your answer.
Finding the Nth Term• Consider the sequence
3, 6, 12, 24, …
• Define it Recursively a1 = 3
a2 = 3(2)
a3 = 3(2)(2)
a4 = 3(2)(2)(2)
an = 2an-1
• Define it Explicitly• an = 3(2)n-1
Write a rule for the nth term of the sequence -8, -12, -18, -27, . . . . Then find a8.
Write a rule for the nth term of the sequence 6, 24, 96, 384, . . . . Then find a7.
a1 = 6
Write a rule for the nth term of the geometric sequence. Then find a8.
A ping pong ball is dropped from a height of 16 feet and it always rebounds ¼ of the distance of the previous fall.
a. What distance does it rebound after the sixth time?
b. What is the total distance the ball travels after the this time?
HW #14.3Pg 628-629 1-31 Odd, 32-40
Chapter 14
14-4 Infinite Geometric Sequences
What is happening to the sum as n gets larger?
1, 1 1
2
n
n
Definition Geometric Sequence
If, in an infinite series, Sn approaches some limit as n becomes very large, that limit is defined to be the sum of the infinite series.
If an infinite series has a sum, it is said to converge or to be convergent
Consider the following two sums, which one will have a finite sum and why?
1
3 2n
n
1
13
2
n
n
This infinite series will have a finite sum since |r| < 1
Theorem 14-6
An infinite geometric series is convergent and thus has a sum if and only if |r| < 1
Theorem 14-7
The sum of an infinite geometric series with |r| < 1 is given by
HW #14.4Pg 632-633 1-21 Odd, 23-27
Chapter 14
14-5 Mathematical Induction
Consider the first n odd numbers: 1, 3, 5, 7, 9, …, 2n - 1
Partial Sum
Value
S1 = 1 1S2 = 1 + 3 4S3 = 1 + 3 +5 9S4 = 1 + 3 +5 + 7 16
… … …Sn =. 1 + 3 +5 + 7 + … (2n – 1) n2
We have reasoned inductively that the sum of the first n odd numbers is n2, but we have not proven that to be true.
In Lesson 14-2 we learned that the rule for the sum of the first n positive integers is:
Statements like these can be proven using a method of proof called mathematical induction.
The Principle of Mathematical Induction
If, Pn is a statement concerning positive integers n and
a) P1 is true,
and
b) assuming Pk is true implies that Pk+1 is true,
then Pn must be true for all positive integers n.
2
1
Prove : (2 1)n
i
n n
HW #14.5aPg 636 1-3
Chapter 14
14-5 Mathematical Induction
HW #14.5bPg 636 5, 6, 10-16
HW #R-14aPg 639-640 1-23
Test Review
1 12 12 2
Prove: Thesumof thefirstntermsof anarithmeticseries is
( ( ) ) ( )n n
n nS a n d a a
1
11
Prove: Thesumof thefirstntermsof a geometric series isn
n
rS a
r
After college George got a job working as an aerospace engineer. His starting salary was $40,000 a year and he got a 3% raise every year after that.
a. How much money did he make the 32nd year he worked there.
b. If he retires after working for 40 years, how much money would he have made total?
1
1
aS
r
1
21A
aS
r
2
S(1 )
(1 )A
rS
r
(1 )
(1 )(1 )A
S rS
r r
(1 )A
SS
r
b. In the diagram, there is one L-shaped region corresponding to each positive integer through n. To calculate the area of the nth such region, calculate the areas of the rectangles A, B, and C shown, in terms of n, and simplify your answer.
(Hint: Note that one side of each of the regions A and B has length 1 + 2 + 3 + … + n –1.)
HW #R-14b Pg 641 1-20
