Ch 9 Sequences, Series, and Probability · 4/10/2014 · IN THIS CHAPTER we will discuss counting...

SECTION 9.1 SEQUENCES AND SERIES 4-10-2014 Word 97-2003

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Ch 9 Sequences, Series, and Probability

Have you ever been to a casino and played blackjack? It is the only game in the casino that you can win based on the Law of large numbers. In the early 1990s a group of math and science majors from the Massachusetts Institute of Technology (MIT) devised a foolproof scheme to win at blackjack. A professor at MIT developed a basic strategy outlined in the figure that is based on the probability of combinations of particular cards being dealt, given certain cards already showing. To play blackjack (also called 21), each person is dealt two cards with the option of taking additional cards. The goal is to get a combination of cards that is worth 21 points (or less) without going over (called a bust). You have to avoid going over 21 or staying too far below 21. All face cards (jacks, queens, and kings) are worth 10 points, and an ace in blackjack is worth either 1 or 11 points. The students used the professor’s strategy along with a card-counting technique to place high bets when there were more high-value cards left in the deck. It is reported that in 1992 the team won $4,000,000 from Las Vegas casinos. The casinos caught on and the students were all banned within 2 years. The 2008 movie “21” was based on this event.


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IN THIS CHAPTER we will discuss counting and probability in addition to three other topics: sequences and mathematical induction, and the binomial theorem.

SECTION 9.1 SEQUENCES AND SERIES 9.2 ARITHMETIC SEQUENCES AND SERIES 9.3 GEOMETRIC SEQUENCES AND SERIES 9.4 MATHEMATICAL INDUCTION 9.5 THE BINOMIAL THEOREM 9.6 COUNTING, PERMUTATIONS, ANDCOMBINATIONS 9.7 PROBABILITY CHAPTER OBJECTIVES • Understand the difference between sequences and series. • Find the general, nth, term of a sequence or series. • Understand the difference between finite and infinite series. • Evaluate a finite series. • Determine if an infinite series converges or diverges. • Prove a mathematical statement using induction. • Use the binomial theorem to expand a binomial raised to a positive integer power. • Understand the difference between permutations and combinations. • Calculate the probability of an event. Page 705


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Page 706

SECTION 9.1 SEQUENCES AND SERIES

SKILLS OBJECTIVES • Find terms of a sequence given the general term. • Look for a pattern in a sequence and find the general term • Apply factorial notation. • Apply recursion formulas. • Use summation (sigma) notation to represent a series. • Evaluate a series.

CONCEPTUAL OBJECTIVES

• Understand the difference between a sequence and a series. • Understand the difference between a finite series and an infinite series.

Sequences The word sequence means an order in which one thing follows another in succession. In mathematics, it means the same thing. For example, if we write 2 3 4 5, 2 , 3 , 4 , 5 ,?x x x x x , what would the next term in the sequence be, the one where the question mark now stands? The answer is 66x .

DEFINITION Sequence A sequence is a function whose domain is a set of positive integers. The function values, or terms of the sequence are written as

1, 2 3 , , , ,na a a a… … Rather than using function notation, sequences are usually written with subscript (or index)

notation, subscripta . A finite sequence has the domain { } 1, 2, 3, ..., n for some positive integer n. An infinite sequence has the domain of all positive integers { } 1, 2, 3, ... . There are times when it is convenient to start the indexing at 0 instead of 1:


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0 1, 2 3, , , , ,na a a a a… … Sometimes a pattern in the sequence can be obtained and the sequence can be written using a general term. In the previous example, 2 3 4 5 6, 2 , 3 , 4 , 5 ,6x x x x x x , … , each term has the same exponent

and coefficient. We can write this sequence as n

na nx= , n = 1. 2. 3. 4. 3, 6 where a is called the general term.

EXAMPLE 1 Finding the Sequence, Given the General Term Find the first four (n = 1, 2, 3, 4) terms of the sequences, given the general term.

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Find the first four terms of the sequence ( )2

1 n

nan−

=

EXAMPLE 2 Finding the General Term, Given Several Terms of the Sequence Find the general term of the sequence, given the first five terms.


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Find the general term of the sequence, given the first five terms.

Parts (b) in both Example 1 and Example 2 are called alternating sequences, because the terms alternate signs (positive and negative). If the odd terms, 1 3 5, , ,...,a a a are negative and the even

terms, 2 4 6, , ,...,a a a are positive, we include ( )1 n− in the general term. If the opposite is true,

and the odd terms are positive and the even terms are negative, we include ( ) 11 n+− in the general term.

Factorial Notation Many important sequences that arise in mathematics involve terms that are defined with products of consecutive positive integers. The products are expressed in factorial notation.

DEFINITION Factorial If n is a positive integer, then n! (stated as “n factorial”) is the product of all positive integers from n down to 1.

( )( )! 1 2 3 2 1 2 and 0! 1 and 1! 1n n n n n= − − ⋅⋅⋅ ≥

= =


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The values of n! for the first six nonnegative integers are

0! = 11! = 12! = 2 • 1 = 23! = 3 2 l = 64! = 4 • 3 • 2 1 = 245! = 5 • 4 • 3 • 2 • 1 = 120

Notice that 4! = 4 • 3 • 2 1 =4 3! . In general, we can apply the formula ( )! 1 !n n n= − . Often the brackets are not used, and the notation ( )! 1 !n n n= − implies calculating the factorial (n - 1)! and then multiplying that quantity by n. For example, to find 6!, we employ the relationship n! = n(n - 1)! and set n = 6:

6! 6 • 5! 6 120 720= = =


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EXAMPLE 3 Finding the Terms of a Sequence Involving Factorials

Find the first four terms of the sequence, given the general term !

n

nxan

=

EXAMPLE 4 Evaluating Expressions with Factorials Evaluate each factorial expression.


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COMMON MISTAKE

• YOUR TURN Evaluate the factorial expressions.


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Recursion Formulas Another way to define a sequence is recursively, or using a recursion formula. The first few terms are listed, and the recursion formula determines the remaining terms based on previous terms. For example, the famous Fibonacci sequence is 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, … . Each term in the Fibonacci sequence is found by adding the previous two terms.

We can define the Fibonacci sequence using a general term:

1 2 2 11, 1, 3n n na a and a a a n− −= = = + ≥ The Fibonacci sequence is found in places we least expect them (for example, pineapples, broccoli, and flowers). The number of petals in a flower is a Fibonacci number. For example, a wild rose has 5 petals, lilies and irises have 3 petals, and daisies have 34, 55, or even 89 petals. The number of spirals in art Italian broccoli is a Fibonacci number (13).


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EXAMPLE 5 Using a Recursion Formula to Find a Sequence

• YOUR T U RN Find the first four terms of the sequence:

Sums and Series When we add the terms in a sequence, the result is a series.

DEFINITION Series

Given the infinite sequence 1, 2 3 , , , ,na a a a… … the sum of all of the terms in the infinite sequence is called an infinite series and is denoted by

1 2 3 na a a a+ + + ⋅⋅⋅+ + ⋅⋅⋅ . and the sum of only the first n terms is called a finite series, or nth partial sum, and is denoted by

1 2 3n nS a a a a= + + + ⋅⋅⋅+


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The capital Greek letter Σ (sigma) corresponds to the capital S in our alphabet. Therefore, we use Σ as a shorthand way to represent a sum (series). For example, the sum of the first five terms of the sequence 1, 4, 9, 16, 25, …., n2,.. can be represented using sigma (or summation) notation:

This is read “the sum as n goes from 1 to 5 of n2.” The letter n is called the index of summation, and often other letters are used instead of n. It is important to note that the sum1 can start at other numbers besides 1. If we wanted the sum of all of the terms in the sequence, we would represent that infinite series using summation notation as


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EXAMPLE 6 Writing a Series Using Sigma Notation Write the following series using sigma notation.


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• YOUR TURN Write the following series using sigma notation.

Now that we are comfortable with sigma (summation) notation, let’s turn our attention to evaluating a series (calculating the sum). You can always evaluate a finite series. However, you cannot always evaluate an infinite series.


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EXAMPLE 7 Evaluating a Finite Series

Study Tip The sum of a finite series always - exists. The sum of an infinite series may or may not exist. Infinite series may or may not have a finite sum. For example, if we keep adding 1 + 1 + 1 + 1 +..., then there is no single real number that the series sums to because the sum continues to grow without bound. However, if we add 0.9 + 0.09 + 0.009 + 0.0009 ± this sum is 0.9999... 0.9= , which is a rational number, and it can be proven that 0.9 1= .


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EXAMPLE 8 Evaluating an Infinite Series, If Possible Evaluate the following infinite series, if possible.


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Solution (b): Expand the series. = 1 + 4 + 9 + 16 + 25-36 + This sum is infinite since it continues to grow without any bound. In part (a) we say that the series converges to and in part (b) the say that the series diverges.

• YOUR TURN Evaluate the following infinite series, if possible.


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Applications The annual sales at Home Depot from 2000 to 2002 can be approximated by the model

2 45.7 9.5 - 1 .6na n n= + , where na , is the yearly sales in billions of dollars 0,1, 2n = .

What does the finite series 2

0

13 nn

a=∑ tell us?

It tells us the average yearly sales over 3 yes


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In Exercises 1-12, write the first four terms of the sequence. Assume n starts at 1.

1. na n=

22. na n=

3. 2 1na n= −

4. nna x=

( )5.

1nna

n=

+


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( )16. n

na

n+

=

27.!

n

nan

=

( )!8.1 !n

nan

=+

( ) 19. 1 n nna x += −


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( ) 1 210. 1 nna n+= −

( )( )( )

111.

1 2

n

nan n

−=

+ +

( )( )

2

2

112.

1n

na

n−

=+


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In Exercises 13-20, find the indicated term of the sequence.

9113. ?2

n

na a = =

( )( ) 19

1 !15. ?

2 !

n

n

na a

n−

= =+


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ln4920. ?n

na e a= =


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In Exercises 2 1—28, write an expression for the nth term of the given sequence.


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In Exercises 29 - 40, simplify the ratio of factorials.

Solutions: 29-40


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In Exercises 41 - 50, write the first four terms of the sequence defined by the recursion formula. Assume the sequence begins at 1.


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In Exercises 51- 64, evaluate the finite series.


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In Exercises 65 - 68, evaluate the infinite series, if possible.

In Exercises 69 - 76, apply sigma notation to write the sum.


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77. Money. Upon graduation Jessica receives a commission from the U.S. Navy to become an officer and a $20,000 signing bonus for selecting aviation. She puts the entire bonus in an account that earns 6% interest compounded monthly. The balance in the account after n months is

Her commitment to the Navy is 6 years. Calculate 72A . What does 72A represent?

78. Money. Dylan sells his car in his freshman year and puts $7,000 in an account that earns 5% interest compounded quarterly. The balance in the account after n quarters is

Calculate 12A . What does 12A , represent?

79. Salary. An attorney is trying to calculate the costs associated with going into private practice. If she hires a paralegal to assist her, she will have to pay the paralegal $20.00 per hour. To be competitive with most firms, she will have to give her paralegal a $2 per hour raise per year. Find a general term of a sequence na , which represents the hourly salary of a paralegal with n years of experience. What will be the paralegal’s salary with 20 years of experience?


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80. TL Salaries. A player in the NFL typically has a career that lasts 3 years. The practice squad makes the league minimum of $275,000 (2004) in the first year, with a $75,000 raise per year. Write the general term of a sequence a,, that represents the salary of an NFL player making the league minimum during his entire career. Assuming n = 1 corresponds to the first year, what does

3

1n

na

=∑ represent?

Ch 9 Sequences, Series, and Probability · 4/10/2014 · IN THIS CHAPTER we will discuss counting...

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