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Discovery Precalculus: A Creative and Connected Approach

176

UNIT 7Sequences and Series

We will start this Unit by recalling, from your work in Unit 1, that a sequence is formally defined as a function with a domain of the positive integers, sometimes in union with 0. Notice that this implies that sequences are discrete functions with distinct range values as opposed to the continuous functions that you have generally worked with in the past.

A sequence can be “self-explained,” that is, defined explicitly, as a function of n, or it can be defined recursively where any next term depends upon a previous term or terms. An example of an explicitly defined sequence is

an 10n 3,

where it is assumed that the domain is the positive integers. On the other hand, an exam-ple of a recursive function is

a1 = 8 and an = 2an 1 3,

where, to find the next term of this sequence, one must substitute the given first term into the sequence.

By convention, a sequence can be symbolized in many ways, such as an, a(n), f (n), {an}, or

. Sequences can be finite or infinite. Last, sequences can be characterized as being arithmetic, geometric, or neither.


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Lesson 7.1: Arithmetic and Geometric Sequences

Definition: An arithmetic sequence is a sequence in which the difference between any two consecutive terms is constant. This constant is called the common difference and is usually symbolized as d.

Exploration 7.1.1: Working With Arithmetic Sequences

1. If the first term of an arithmetic sequence is 4 and the difference between con-secutive terms is 5, what is the 2nd term? The third term? The twelfth? The eighti-eth? The nth?

2. An explicit sequence is given by an 33 2n. Is this sequence arithmetic? How can you tell? Graph this explicit sequence using a domain of positive integers n from 1 to 10 inclusive.

3. How much information does one need to determine an arithmetic sequence? That is, if a person tells you that she is thinking of an arithmetic sequence, how many, or which, terms must be revealed such that you will be able to create the function that generates the terms of the sequence?

4. Write a recursive formula for the sequence: 7, 17, 27, 37, 47, …

5. Write an explicit formula for the sequence: 1, 2, 4, 8, 16, 32, …

In order to find a specific term value of an arithmetic sequence, one can use the formula

an a1 (n 1)d

6. Find the 6th term of the arithmetic sequence that has a1 12 and a2 20. Find the 50th term of the above sequence.


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Exploration 7.1.2: Working With Geometric Sequences

1. If the first term of a geometric sequence is 3 and the common ratio between terms is 2, what is the 2nd term? The third term? The twelfth? The eightieth? The nth?

2. How much information does one need to determine a geometric sequence? That is, if a person tells you that she is thinking of a geometric sequence, how many, or which, terms must be revealed such that you will be able to create the func-tion that generates the terms of the sequence?

3. Find the 7th term of the geometric sequence which has a1 2 and a2 8. Find the 20th term of the above sequence.

Geometric Sequences and Series

Definition: A geometric sequence is a sequence in which the ratio between any two consecutive terms is the same. This common ratio is usually symbolized as r.

To find any term of a geometric sequence, one can use the formula

an a1 r n 1


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Exploration 7.1.3: Arithmetic or Geometric?

Fill in the table:

Sequence Arithmetic or Geometric? d or r Find the 20th

Termnth Term Formula

–7, –3, 1, …1, 4, 9, …81, 54, 36, …25, 50, 100, …1, 2, 6, 24, …

As stated previously, not all sequences are arithmetic or geometric. A famous sequence that is neither arithmetic nor geometric is the Fibonacci Sequence which can be written as 1, 1, 2, 3, 5, 8, 13, 21, 34, … You might wish to do some research on the Fibonacci Sequence for it has a rich history and many connections to other topics in mathematics. For example, most of you have probably heard of Pascal’s Triangle. Can you find the Fibonacci Sequence within Pascal’s Triangle?

Quick Exercise:

1. How is the Fibonacci Sequence generated?

2. Can you create three more sequences that are neither arithmetic nor geometric?


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Lesson 7.2: Convergent Sequences

Definition: A sequence {an} converges to a number c, if {an} stays arbitrarily close to c as n gets very large. Concerning notation, the statement that “{an} converges to c” is often stated as “{an} has limit c” and is written as “ lim

n→∞an = c ”.

Exploration 7.2.1: Convergent Sequences

1. Which of the following sequences converges to a limit? If the sequence converges, what does it converge to? If not, why doesn’t it converge?

a. 3, 7, 11, …b. ½, 2/3, ¾, 4/5, …c. 1, 1, 1, 1, …d. 2, 3, 2, 3, …e. 1, 1 ½, 1 ½ ¼, …

2. Construct a sequence where…

a. the first term is 6 and it converges to 8.b. the terms are all between 1 and 1 but the sequence doesn’t converge.


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Lesson 7.3: Series and Partial Sums

Definition: A series is the indicated sum of the terms of a sequence. A series can be finite or infinite.

A series Sn can be symbolized using summation notation

Sn = ank=1

n

∑ .

This is read “the series Sn that is the sum of the first n terms of the sequence an.”

The nth partial sum of a series is the sum of the first n terms of that series. The symbol Sn is also thought of as the nth partial sum of a series.

Consider the series 3 5 7 9 … Note that S1 3, S2 8, S3 15, S4 24

This situation could also be written

S4 = 2k +1k=1

4

∑ .

Exploration 7.3.1: Series, Partial Sums, and More

Part 1

FInd:

1. 2 ⋅3kk=1

3

∑ 2. −1( )k 3k − 2( )k=1

6

∑ 3. −1( )k−1 2( ) 3k−1( )k=1

5

∑

Write using ∑ notation:

4. S10 for 1 + 4 + 9 + 16 + 25 + ... 5. S100 for 13+ 14+ 15+ 16+ 17+ ...

6. S50 for 3 + 6 + 12 + 24 + 48 + ....


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Part 2

Definition: An arithmetic or geometric series is a series which results from adding the terms of an arithmetic or geometric sequence, respectively.

Given a finite arithmetic series, its sum can be found by

Sn =n2a1 + an( )

7. Find the sum of the arithmetic series 1 + 3 + 5 + 7 + 9 + ... + 145.

An alternate formula for finding the sum of a finite arithmetic series is

Sn =n22a1 + n −1( )d( )

where d is the common difference between the terms of the series.

[Challenge: Can you derive this formula from the previous one given?]

8. Find the 100th partial sum of the arithmetic series with a1 = 17 and d = 4 .

9. Find 5k + 2k=1

30

∑ =

Part 3The sum of a finite geometric series can be found by

Sn =a1 1− r

n( )1− r

10. Derive the formula for the sum of a finite geometric series.

[Hint: a. Start with the general geometric series

Sn = a1 + a1r + a1r2 + a1r

3 + ..+ a1rn−1

with common ratio r

b. Multiply both sides of this equation by -r.

c. Add the results of a. and b. and then do some algebraic rearranging.]

11. Find S10 for the geometric series with a1 = 5 and r = 3 .


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Lesson 7.4: Convergent Geometric Series

Definition: An infinite geometric series converges to a number s if the partial sums, Sn, stay arbitrarily close to s as n gets very large.

Exploration 7.4.1: Convergent Geometric Series

Consider the formula for the sum of a geometric series

.

1. What will happen to the formula above if r is such that r < 1 and n gets very large?

2. Use your answer to (1.) to justify the following statement:

A geometric series converges to the finite number s, if r < 1. The number s can be found by

s = limnSn =

a11 r

3. To what number does the geometric series 1+ 12

+ 14

+ 18

+ ... converge?

4. Find the sum of an infinite geometric series if a1 = 5 and r = 13

.

Sn a1(1 – r n)

1 – r


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Definition. A decimal representation for a nonnegative real number is an expression of the form N.a1a2a3… where N is a nonnegative integer and ai are integers between 0 and 9.

Exploration 7.4.2: Series Representations of Repeating Decimals

1. Use fractions to write 0.11111… as a series and verify that this is an infinite geometric series.

a. What is the value of r for this geometric series?

b. Use this information to find the fraction to which this infinite geometric series converges.

2. Use fractions to write 0.99999… as a series and verify that this is an infinite geometric series.

a. What is the value of r for this geometric series?

b. Use this information to find the fraction to which this infinite geometric series converges.

c. Is this result surprising to you? Explain.


185

Exploration 7.4.3: Applications of Geometric Series

1. For a certain drug to have the desired healing effect and be safe, the amount in a person’s system should be between 18 and 22 micrograms. After 6 hours the amount of a dosage of the drug left in the body is 70% of the original dosage. Each dosage of the drug delivers 6 micrograms to the body.

a. Assume a person receives one dosage of the drug every six hours. Write a geometric series representing the amount of the drug in a person’s system after n dosages.

b. When (if ever) would the amount of the drug in a person’s system be greater than 18 micrograms?

c. Will the amount of the drug in a person’s system ever be greater than 22 mi-crograms?

2. Suppose one drops a rubber ball from a height of 20 feet above the floor. After each bounce, the ball rebounds to 80% of the maximum height of the previous bounce.

a. Draw a picture of this situation showing the height of the ball after each bounce over time.

b. Write the first few terms of the geometric series that represents the total distance the ball travels (both up and down) after each consecutive bounce of the ball.

c. What total distance has the ball traveled (up and down) after 10 bounces [Don’t forget to include the initial drop height of the ball.

d. Suppose the ball makes an infinite number of bounces before coming to rest. Will the ball travel an infinite distance up and down or will the total distance traveled be finite? If the distance is finite, what would it be?


186

Lesson 7.5: Induction

The use of mathematical induction to prove statements about infinite series is an important tool in mathematics. This last Lesson allows one the chance to work with mathematical induction as a technique for proving some mathematical statements.

The concept behind mathematical induction is that, when given a statement about the natural numbers, we can establish the validity of the statement by following a certain line of logic. First we show that the statement works for an initial natural number value (usually n 1). Then we assume that the statement works for the first k natural numbers. Last, we show that the statement is true for the (k + 1)th natural number, no matter what k is. One way to envision why this works is to picture a row of dominoes. Set up correctly, if you can make the first k dominoes fall by knocking the first one down, then you can be assured that all the dominoes will fall.

DEFINITION: The Principle of Mathematical Induction. Let P(n) be a statement that is either true or false for each n . Then P(n) is true for all n provided that

a. P(1) is true, [THE BASE CASE] and

b. for each n , if P(k) is true [THE INDUCTIVE HYPOTHESIS], thenP k +1( ) is true.

Exploration 7.5.1: Mathematical Induction

1. Suppose P(n) = 1 + 2 + 3 + ... + n. Prove that 1 + 2 + 3 + … + n = 1 ( 1)2

n n + for

every natural number n. [Hint: You will do this by completing a c. below.]

a. Show that the base case P(1) is true.

b. Assume that P(k) is true and rewrite the statement to be proven from (1).

c. Now, write the statement of P(k 1) and show that the statement is true.

2. Prove 1 3 5 7 … (2n 1) = n² for every natural number n.

3. CHALLENGE: Can you extend your knowledge of how to use mathematical in-duction in order to prove that 7n – 4n is a multiple of 3, for all n ?


187

Lesson 7.6: Probability and Combinatorics - a Momentary Di-version

In this lesson, we will allow ourselves a momentary diversion into the area of probability and, specifically, the field of combinatorics in mathematics. We introduce useful count-ing techniques that are pertinent to probability theory. Our ultimate goal, however, is to develop the theory needed to explore our ultimate topic of interest, the binomial theorem. The binomial theorem will be explored at length in the next lesson.

At this point in your mathematical careers, it is quite likely that you have had some expe-rience with calculating the probability that certain equiprobable outcomes occur within finite sample spaces. The probability of such an event occurring consists of the number of events constituting the event divided by the total number of outcomes in the sample space. While this counting method may sound fairly straightforward, it is often the case that actu-ally calculating the way certain outcomes can occur can be quite an unruly process.

In order to perform these types of calculations in probability, we appeal to the field of com-binatorics.

Definition. A study of different possibilities for arrangements of objects is known as combinatorics.

Essential to such a study is the rule known as the counting principle.

Definition. The counting principle states that if one event can be chosen in m different ways, and another event can be chosen in n different ways, then the two events can be chosen in m . n different ways.

The counting principle can be extended to any number of independent events. We focus on an extension of the counting principle known as a combination.

Definition. A combination is an arrangement of objects without repetition where order in not important. The number of combinations of n objects chosen r at a time is sym-bolized C(n,r) where

C(n,r) = n!(n r)!r!

.


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Recall that n! = n(n – 1)(n – 2)∙ ... ∙2∙1. Also note that 0! 1 and 1! 1. Last, we point out that the expression C(n, r) can also be denoted as

nr( (.

DEFINITION A permutation consists of different arrangements of objects where the order of the objects is important. The number of permutations for n objects taken n at a time is

P n,n( ) = n!

Furthermore, the number of permutations of n objects taken r at a time is

P n,r( ) = n!n − r( )!

.

Exploration 7.6.1: Counting

Part 1

1. Suppose one conducts an experiment where a die is rolled and then a coin is flipped. Each outcome pair is recorded. For example, one possible outcome pair is 2H, where 2 is the number that is displayed after rolling a die and H stands for the coin flip that resulted in a “heads.” Use the counting principle to find the number of outcome pairs that are possible for this experiment. A tree diagram may be used to help visualize your result.

2. A pizza shop offers 3 different kinds of meat and 5 different kinds of vegetables as possible cheese pizza toppings. How many different choices does a customer have if she wants 1 meat topping and 1 vegetable topping on her cheese pizza? A tree diagram may be used to help visualize your result.


189

3. An automobile license plate usually has 3 letters of the alphabet on it along with 3 numbers; each number belongs to the set {0, 1, 2, 3,…, 9}. These letters or numbers can be repeated on the plate. Use this information to write an equa-tion that will find N, the total number of combinations of different license plates possible.

Part 2

Use what you’ve learned about combinations to answer the following questions.

4. From a list of 10 different songs, how many 5-song playlists can be created?

5. How many different groups of 5 basketball players can be selected from a team of 10 players?

6. How many different ways are there to choose a president, vice-president, and secretary from a group of 20?

7. A track and field relay team consists of 4 team members lined up in order around a track in the 4 x 100 meter relay. How many different line-ups are pos-sible if the 4-man relay team is chosen from among 6 possible team members?

8. Show that

C n,r( ) = P n,r( )r!

.


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Lesson 7.7: The Binomial Theorem

Consider the expansion of the binomial powers (x + y)n, as listed in Figure 7.7-1.

Figure 7.7-1: A Sample of Binomial Expansions for n 0, 1, 2, 3, 4, 5.

Based on previous experience, you might have already noticed that the binomial coefficients of the expansions above are the same as the array known as Pascal’s Triangle. Pascal’s triangular array contains many interesting properties and patterns, some of which will be explored in this lesson. The first few rows of Pascal’s Triangle are listed in Figure 7.7-2.

Figure 7.7-2: Pascal’s Triangle for Rows 0, 1, 2, 3, 4, 5.

x + y =1

x + y =1x +1y

x + y =1x + 2xy +1y

x + y =1x + 3x y + 3xy +1y

x + y =1x + 4x y + 6x y + 4xy +1y

x + y =1x + 5x y +10x y +10x y + 5xy +1y

0

1

2 2 2

3 3 2 2 3

4 4 3 2 2 3 4

5 5 4 3 2 2 3 4 5

( )

( )

( )

( )

( )

( )

Row 0

Row 1

Row 2

Row 3

Row 4

Row 5

11 1

1 2 11 3 3 1

1 4 6 4 11 5 10 10 5 1

Row 0

Row 1

Row 2

Row 3

Row 4

Row 5


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In fact, the binomial coefficients are simply an array of combinations of the form

.

This is also true of the Pascal’s Triangle array as displayed in Figure 7.7-3.

Figure 7.7-3: Pascal’s Triangle – An Array of Combinations

We define the binomial theorem:

Definition. The Binomial Theorem. For all positive integers n and x, y belonging to the real numbers

x + y( )n = n0xny0 + n

1xn 1y1 + n

2xn 2y2 + ...+ n

n 1x1yn 1 + n

nx0yn

= nkk=0

nxn kyk

( ( ( ( ( ( ( (Σ ( (

( (

nr( (k


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The question is, “Why does the binomial expansion of (x y)n for various powers of n lead to an array of combinations?” An informal justification of this fact starts with noticing that

x + y( )n = x + y( ) x + y( )... x + y( )n times

! "### $### .

It is also worth noting that each term of the distributed expression of (x y)n has the form x r y s where r and s are positive integers such that r s n. One of x or y is chosen for each of the n factors of (x y). Thus each term x n k y k in the binomial expansion is obtained by

choosing y from any k of the n factors. Therefore, there are kr

⎛⎝⎜

⎞⎠⎟

ways to select k items

from the entire set of n items, which explains the reason for nk

⎛⎝⎜

⎞⎠⎟

the binomial coefficients in the binomial expansion.

Exploration 7.7.1: Binomial Expansion

1. What coefficient will yield the coefficient of the term x10 y 5 in the expansion of (x y)15 ?

2. Find the fifth term of the expansion of (a b)11.

3. Expand (5x y)6 using the binomial theorem.

4. Expand (3 i)5, where i is the imaginary number, using the binomial theorem.


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Exploration 7.7.2: Pascal’s Triangle Properties

1. Add the numbers along each diagonal in Figure 7.7.2-1. Do the numbers look familiar? Explain.

Figure 7.7.2-1

2. In Pascal’s Triangle, add the numbers across each row and create a sequence of these numbers. Do you notice a pattern in this sequence?

3. Verify the following two identities and describe how they relate to Pascal’s Triangle: For all r ≤ n,

a. nr

⎛⎝⎜

⎞⎠⎟=

nn − r

⎛⎝⎜

⎞⎠⎟

b.

4. Research Sierpinski’s Triangle and explain the relationship between this triangle and Pascal’s Triangle.

nr −1

⎛⎝⎜

⎞⎠⎟+

nr

⎛⎝⎜

⎞⎠⎟=

n +1r

⎛⎝⎜

⎞⎠⎟


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Conclusion

It is the authors’ sincere hope that the Explorations contained within this text have enticed you to think deeply about some of the mathematics you’ve encountered previously, about new ideas presented, and about the connections between the two. It is also our hope that the inquiry-based “open forum” methodology used to deliver this course has broadened your perspective and encouraged you to consider pursuing mathematics as a major in the future.

Best wishes,

The Authors

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