module 8: using recursively- defined functions to...

Module 8: Using Recursively-Defined Functions to Module

and Solve Problems Prepared by Dr. Cos Fi, UNC Greensboro Stephanie Gallop, Guilford County Schools

Chapter

8.1. North Carolina Standard Course of Study ................................................................................210

8.2. vocabulary...............................................................210

8.3. TEXTBOOKS................................................................210

8.4. SEQUENCES ...............................................................211

8.5. ACTIVITIES (HANDSHAKES).......................212

8.6. MAKING CONNECTIONS..................................213

8.7. ACTIVITIES (PERMUTATIONS)..................215

8.8. MAKING CONNECTIONS...................................217

8.9. TYPES OF SEQUENCES...............................219

8.10. DEFINITION......................................................................220

8.11. GEOMETRIC SEQUENCE................................221

8.12. ACTIVITIES.....................................................................223

8.13 EXTENSION...................................................................227

8.14 CONVERGENCE & DIVERGENCE OF SEQUENCES................................................................228

8.15 SERIES..................................................................................231

8.16 SUMMATION NOTATION ..................................232

8.17 SERIES CONVERGENCE.................................234

8.18. FINITE DIFFERENCES.......................................236

8.19 ACTIVITIES......................................................................237

8.20 ADDITIONAL RESOURCES...............................240

8

209 Sponsored by NC Math and Science Education Network

Advanced Functions and Modeling Workshop Summer 2004

8.1. North Carolina Standard Course of Study

GOAL 2: The learner will use functions to solve problems.

2.05 Use recursively-defined functions to model and solve problems.

a) Find the sum of a finite sequence. b) Find the sum of an infinite sequence. c) Determine if a given series converges or diverges. d) Translate between recursive and explicit representations.

8.2. Vocabulary

Linear Sequence Geometric Sequence Geometric Series Subscript Notation Summation Notation

Converge Diverge Translate between Recursive and Explicit Representations

8.3. Textbooks:

Holiday, B., Cuevas, G., McClure, M., Carter, J., Marks, D. (2004). Advanced Mathematical Concepts: Pre-calculus with Applications. Columbus, OH: Glencoe.

(Chapter 11)

Connally, Hughes-Hallett, Gleason, et al. (2004). Functions Modeling Change: A Preparation for Calculus. USA: John Wiley & Sons, Inc.

(Chapter 12)

Materials: Graphing Calculators Tower of Hanoi physical model Graphing paper General Math Supplies

Module 8: Recursively Defined Functions 210


8.4. Sequences

DEFINITION/IDEA/DISCUSSION

A sequence is an ordered list or progression of numbers. Entries in the list (called a sequence) are called terms of the sequence. So we have the

first term, second term, third term, fourth term, fifth term, sixth term, seventh term, eighth term, ninth term, tenth term, eleventh term, twelfth term, thirteenth term, fourteenth term, …, fortieth term, …, nth term, …,

and so on depending on the size of the sequence.

Some examples of sequences With which you are already familiar are:

1) The counting or natural numbers: 1, 2, 3, 4, 5, 6, 7, … . Here the first term is 1, the second term is 2, the third terms is 3, the nth term is n, where n is a natural number

2) The set of the following positive even numbers: 2, 4, 6, 8, 10, 12, and so on. Here the first term is 2, the second term is 4, the third term is 6, the fourth term is 8, the nth term is 2n. Why? Note that n is a natural number.

3) The set of positive odd numbers: 1, 3, 5, 7, 9, 11, 13, …, and so on. Yes, you got it. The first term is 1, the second term is 3, the third term is 5, the fourth term is 7. What is the nth term? Describe how you came up with the nth term.

4) Can you think of another sequence that you already know? Write it down. Share with a partner. Did your partner come up with the same sequence?

Teacher Notes:

Introduce students to the idea of developing/maintaining a mathematical toolkit where they can record definitions, ideas, concepts, procedures, etc. The toolkit should be separate from their regular notebook. You may choose to let students use their toolkit on assessments. You may choose to collect their toolkits and grade them.

Possible answers include: Sequence of primes, squares, cubes, multiples of numbers, powers of natural numbers, etc.

This will be a good place to emphasize that any ordered list of numbers is called a sequence and that there are infinitely many ways of ordering numbers. Hence there are infinitely many sequences. Have students write their different sequences on the board and have them copy the set of sequences that the class generated into their tool-kit.



Name: Period: Group/Whole Class:

8.5. Activities:

(1) How many handshakes are possible in your class? In this activity, you will use mathematics to model number of handshakes among any given number of people. To do this, you will physically shake hands and count the number of the handshakes. To begin the process we will look at a scaled down bite-size of the problem, and generate a pattern that can be used to predict the number of handshakes among any number of people. The approach of looking at a simpler situation is a mathematical tool for conquering big problems.

(a) Write down the number of students in your classroom. Include yourself in the count.

(b) How many handshakes are possible if there is only one student?

(c) How many handshakes are possible between two students?

(d) How many handshakes are possible among three people?

(e) How many handshakes are possible among four people?

(f) Let us organize the data into a t-table. In the first column write number of students, in the second column have students write number of handshakes. Continue until you have answered the question posed in (1).

Number of Students Number of Handshakes

(g) Is there is a pattern that relates the number of handshakes to the number of students?




8.6. Making Connections:

In these three activities we will investigate the same mathematics ideas/concepts in different situations.

A. Circle – Points – Chords Activity Draw a circle. Place a point on the circle. How many chords can you draw between the point?

Now include a second point on the circle. How many connecting chords are there between the two points?

Continue adding points, until you have generated a pattern for the number of line segments among any number of points on a circle. Organize the data as in the previous activity.

Number of Points Number of Connecting Segments

What is the pattern? How does the circle and chords investigate compare to the handshake problem? Suppose we use the points to represent students, how many points will you need to put on the circle and how many chords will you have connecting the points?

B. How many 2-member committees (partners) are possible if you have 2 people, three people, four people, five people, …, n people?

C. Investigate triangular numbers and make connections to the previous activities.



Teacher Notes

Highlight the beauty of sequences as a capturing device for patterns, with the possibility of making predictions. Help students represent the sequences in list form with their labels. Ask students to place the lists in their tool-kit for use in the future.

You should also point out that we can use a shorthand for a sequence such as to represent the sequence ,

where is the first term of the sequence, …, is the nth term.

Formally, sequences are defined as functions on the positive integers or subsets of positive integers. So, the positive integers form the domain for sequences. Another way to think about this is that the positive integers form the index set for sequences. The idea of index will become very important when we discuss series.

an{ } a1, a2, a3, a4 , K, an , K

a1 an




8.7. Activities:

(2) Permutations In this activity you will work in pairs. Your teacher will present your group with 5 different objects: books, pencils, etc. Your task is to arrange the different objects in different orders starting with one of the objects. Now arrange two of the objects. Arrange three of the objects. Arrange four of the objects. Finally, arrange all five of the objects.

(a) List the number of ways.

Number of Objects Number of Distinct Ordered Arrangements

(b) Write the ordered arrangements as a progression

(c) Predict the next term of the sequence

(d) Explain your reasoning

(e) Now see if you can predict the next two terms.



Teacher Notes

Formalize this process with the factorial sequence. Can also consider a sequence of the reciprocals of the factorial sequences, starting with 0!. We will call on the sequence of the reciprocals of factorials when we discuss series to generate Euler number e. See pages 807 – 808, and example 2.




8.8. Making Connections:

Write down the number of students in your class (include yourself).

A. Single Assignment of Lockers

How many ways can you assign one student to one locker if you wanted to assign every student in your class a locker?

B. Double assignment of two students to one locker

How many ways can you assign the lockers to every student in your class?

C. Card shuffle

How many different shuffles are possible in a deck of 52 cards? Is it possible for a human being to produce all of those shuffles in her life time?



Teacher Notes

At this point students would have gotten the idea of a sequence as a pattern/order of numbers that has useful characteristics such as a way of organizing information so that patterns are rendered visible and can be manipulated to support predictions and explanations.



8.9. Types of Sequences

(1) Linear/Arithmetic Sequence (Section12 –1 pages 759 – 765)

A sequence of numbers with a constant difference between any two consecutive terms

Natural numbers, evens, odds, and multiples from the previous exercises are arithmetic. At this point you will learn how to define arithmetic sequences symbolically. That is, we will move beyond a progression/list representation of arithmetic sequences towards an explicit definition. We will also write the arithmetic sequence recursively.

If all the ?’s are equal, then the sequence is arithmetic.

a1,a2,a3,a4 ,a5,a6,K,an,an+1K

a2 − a1 = ?

a3 − a2 = ?

...

an+1 − an = ?

And we can write: , for all n. The d is called the constant difference. Furthermore, we can represent the arithmetic sequence as follows

Generate representations for the fifth, sixth, and seventh terms.

Use the patterns to come up with the nth term given by

an+1 − an = d

a1

a2 = a1( )+ d

a3 = a2( )+ d = a1 + d( )+ d = a1 + 2d

a4 = a3( )+ d = a1 + 2d( )+ d = a1 + 3d

...

an = ...

an = a1 + n −1( )d . This is the explicit definition of arithmetic sequences.



8.10. Definition/Idea

Recursive definition of a sequence is a process of describing present and future terms of the sequence using previous terms of the sequence. The recursive definition of a sequence has two components:

a) A starting point or few starting points (usually the first term or the first and second terms)

b) The recursion formula (used to build the sequence from the starting points)

In the case of arithmetic sequence (A) The starting point is (B) The recursion formula is

a1 an = an−1 + d , for n ≥ 2.

You might also see the recursive definition of an arithmetic sequence as

Look at the sequences you have so far in your tool-kit and identify those that are arithmetic sequences.

Why do you think they are arithmetic?

What is the constant difference in each of the sequences you classified as arithmetic?

Write both the explicit and recursive definitions of your arithmetic sequences.

***Investigate more questions from the textbook on pages 763 – 765 on arithmetic sequences

{ } 1 1

1 , 2nn n

a aa

a a d n−

=⎧= ⎨ = + ≥⎩




(2) Geometric Sequence (section 12 – 2, pages 766 – 773)

A sequence of numbers with a constant ratio between any two consecutive terms

1, 2, 4, 8, 16, … ; 3, 9, 27, 81, …;

1,

12

,14

,18

,1

16,K

are geometric sequences.

Let us see why the above sequences are geometric.

a1,a2,a3,a4 ,a5,a6,K,an,an+1K

a2

a1

= ?

a3

a2

= ?

...

an+1

an

= ?

If all the ?’s are equal, then the sequence is geometric.

And we can write: an+1

an

= r , for all n. The r is called the constant ratio. Furthermore, we can re-

present the geometric sequence as follows

a1

a2

a1

= r

a2 = a1( )ra3 = a2( )r = a1r( )r = a1r

2

a4 = a3( )r = a1r2( )r = a1r

3

...

an = ...

Generate representations for the fifth, sixth, and seventh terms.




= n−Use the patterns to come up with the nth term of a geometric sequence which is given by

1. This is the explicit definition of geometric sequences. an a1r

Make sure this makes sense to you.

You can also define geometric sequences recursively as an = an−1r , with the starting point as the first term.

an{ }=a1 =

an = an−1r, n ≥ 2

⎧ ⎨ ⎩

Practice writing both recursive and explicit definitions of geometric sequences.

***Investigate questions from the textbook on pages 772 – 773 on geometric sequences




8.12. Activities:

1. Populations

A. In a given year, say 2000, Midville had a population of 60,000. Demographers predicted in 2000, that the population of Midville will grow at about 2% each year for the next 10 years.

Staring with year 2000, generate a sequence of the population of Midville through 2010.

What do you notice about the sequence? Is the sequence of population values arithmetic, geometric, or neither? Why?

B. Another social scientist projected that after 2010, Midville will gain about 200 people for another decade. Model this situation with a sequence. Identify the type of sequence. How is the population change in this decade different from the previous decade?

C. Due to unforeseen circumstances, Midville lost population from 2020 to 2025 at a rate of half a percent each year – something about a kryptonite explosion. However, it gained population for another decade at a rate of 1.5%. What is the population of Midville at the end of this period?

D. Provide a visual representation of the population of Midville from 2000 to 2035. What is the net change in population? What is the total change in population?

E. Can you forecast the population change for Main City for the next decade? What assumptions did you make? How did you come to your assumptions?




2. Money A. Compound Interest and Amounts

Simple Interest

The simple interest I that accrues on a deposit (principal P) in a bank account can be found by I P r t= × × , where r is the interest rate per annum, and t is the time in number of years. The rate r is a percentage value, hence it is written in decimal representation in the calculating the accrued interest I.

Teacher Note

Work out some problems with students (deploy the following diagrammatic representation of simple interest calculations.

see textbook: example 2, page 816). You can

trP

I

The amount A you get when you add the accrued interest I to the principal P is given by A = P + I. Since I = Prt, the amount A = P + Prt = P(1+rt).

The balance, after interest is added to the principal is, A = P(1 + rt)

Now suppose you want to take interest on the balance, and generate a new balance after adding the new accrued interest to the previous balance, what will you do?

Anew = Aold + Inew , where Inew is the interest on the old balance.

If we represent (old A) as

Anew = Aold + Inew

Anew = Aold + Aold rt( )Anew = Aold 1+ rt( )Anew = P 1+ rt( ) 1+ rt( )= P 1+ rt( )2

A1, and (new A) as A2, then we can compute



A3 = A2 + IA2

A3 = A2 + A2 rt( )A3 = A2 1+ rt( )A3 = P 1+ rt( )2

1+ rt( )A3 = P 1+ rt( )3

What we have just shown is the process of compounding the interest on the principal or balance on a yearly basis.

Generate the 4th, 5th, and 10th balances in the sequence of annual balances when you compound annually. Generate the general case.

In general, when you compound the interest on a yearly basis, over n number of years, the balance you get will be An = P 1+ r( )n

Now suppose we want to compound at different lengths of time, not just per annum, then we will have to generate a new interest rate that applies to the new length of compounding. Suppose there are k periods in one year that we want to compound the interest. Then our new interest for the compounding will be

rnew =Annual rate

# of periods in one year=

rk

.

The number of compounding periods in t number of years will be kt.

If we follow our previous logic, we will generate Akt = P 1+rk

⎛ ⎝ ⎜

⎞ ⎠ ⎟

kt

.

Choose a starting principal (P), annual rate (r), and length of years (t). Have students explore cases when

annual or yearly semi-annual bi-monthly monthly

daily, and

1k = 2k =6k =12k =365k = 12

k = bi-annual or bi-yearly

The values for each of the cases form a sequence.

See pages 769 – 770, example 6 Page 771, number 15 Page 772 – 773, (Numbers 43, 47, 49)




(3) Fibonacci Sequences (See section 12 – 7, pages 806 – 814) The Fibonacci sequence is attributed to Leonardo Pisano of Italy. He wrote about this sequence in his book Liber Abaci in the year 1202. He came upon this sequence by examining the following situation:

1 Start with a pair of rabbits. 2 Follow the following schedule for breeding the rabbits. Each month a pair of

rabbits breeds another pair of rabbits. But the newly bred pair of rabbits does not breed until a month after they have been bred. Said another way, rabbits that are bred in November will start breeding a pair a month in January.

(1) and (2) above provide us with a rudimentary description of the recursive process for defining the Fibonacci sequence. The following table captures the description with numbers and shows the progression of the number of pairs of rabbits. Let us assume we started with a pair of rabbis that began breeding in November

Month Number of Pairs of Rabbit Bred in the month November 1

December 1

January 2 (from the original pair and the pair that was bred in November)

February 3 (from the original pair, the November pair, and the December pair)

March 5 (why?)

April 8 (why?)

Use the pattern in the previous table to complete the following table for the number of pairs of rabbits bred in the following months.

Month Number of Pairs of Rabbits bred in the month

May N = ?

June N = ?

July N = ?

August N = ?

September N = ?

October N = ?

The recursive definition of the Fibonacci sequence is




Verify the numbers you generated for May through October.

1

2

1 2

11

, where 3 monthsn n n

FFibonnaci F

F F F n− −

=⎧⎪= =⎨⎪ = + ≥⎩

8.13 Extension What are the total numbers of pairs of rabbits at the end of each month? The totals form another sequence of Fibonacci numbers, albeit the starting points are different.

Write the sequence here.

A related sequence is the Lucas numbers:

How is the Lucas sequence similar or different from the Fibonacci sequence?

The explicit definition of the terms of the Fibonacci sequence is a little complex and involves Golden Ratios. Use the following definition to practice your algebra (substitution) in verifying the Fibonacci numbers:

1, 3, 4, 7, K

Fn =

1+ 52

⎛

⎝ ⎜

⎞

⎠ ⎟

n

−1− 5

2

⎛

⎝ ⎜

⎞

⎠ ⎟

n

5

Verify the Fibonacci numbers:

Also see pages 812 – 813, numbers 38, 40, 41



8.14. Convergence and divergence of sequences A finite sequence (a sequence with finite number of terms) is always convergent because it has a limit. Infinite sequences, on the other hand, may be convergent or divergent, depending on whether exists. The question is, “Do the terms cluster around a distinct finite value as you move along the progression?” If the terms do not cluster around or attain a distinct finite value, then the sequence is divergent. Another way of saying this is:

If

limn→∞

an

limn→∞

an = L, then the sequence is convergent. (This can be concretized with a graphical representation)

Use some questions from the textbook to investigate the ideas of convergent and divergent sequences. You can discuss sequences in the students toolkits as starters.

Extras 1. Explore the convergence of arithmetic sequences 2. Explore the convergence of geometric sequences



Teacher Note

Extension Continuous Compounding [This does not produce a sequence]

Suppose we want to compound continuously, then the compounding information changes to account for the continuous nature of the event. Basically we would like to know the behavior of

1+rk

⎛ ⎝ ⎜

⎞ ⎠ ⎟

k

A = P 1

as k approaches infinity. Another way to ask the question is, how will the behavior of

+rk

⎛ ⎝ ⎜

⎞ ⎠ ⎟

kt

investigate the lim

be affected as k approaches infinity. To find out, we will use calculus to

it of 1+rk

⎛ ⎝ ⎜

⎞ ⎠ ⎟

k

. As k approaches infinity, the expression is of the form 1∞. So if

we manipulate the expression a little bit and get it into 00

or ∞∞

, then we can apply l’Hôpital’s

Rule.

Using , we can rewrite f x( )[ ]g x( )= e

ln f x( )[ ]g x( )( )= eg x( )ln f x( )( )

limk→∞

1+rk

⎛ ⎝ ⎜

⎞ ⎠ ⎟

k

= limk→∞

ek ln 1+

1k

⎛ ⎝ ⎜

⎞ ⎠ ⎟ .

With further manipulation we get

limk→∞

k ln 1+rk

⎛ ⎝ ⎜

⎞ ⎠ ⎟ = lim

k→∞

ln 1+rk

⎛ ⎝ ⎜

⎞ ⎠ ⎟

1k

,

which is of the indeterminate form 00

. So by applying l’Hôpital’s Rule (using the quotient of the

derivatives), we get

limk→∞

−rk 2

1

1+rk

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

−1k 2

= limk→∞

r

1+1k

⎛ ⎝ ⎜

⎞ ⎠ ⎟

= r .

Hence

limk→∞

1+rk

⎛ ⎝ ⎜

⎞ ⎠ ⎟

k

= limk→∞

ek ln 1+

1k

⎛ ⎝ ⎜

⎞ ⎠ ⎟ = .

So

er

limPk→∞

1+1k

⎛ ⎝ ⎜

⎞ ⎠ ⎟

k⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

t

= P er( )t= Pert

What we have shown is that when we compound interest continuously, the balance after interest is A = rt . Pe



Choose some exercises from the book for students to explore. Choose compound interest and annuities problems. You should pull in ideas from prior work on exponential growth and decay in here and revisit some of the concepts of continuous variables.

-- See page 715 number 11



8.15. Series

A series is a sum of the terms of a sequence ( ). If there are finite number of terms in the series, then the series (in this case, the series is a finite series) will be convergent. If there are an infinite number of terms in the sequences that you are adding, then the series may converge or diverge. To get a better handle on infinite series (that is, there are infinite number of terms that you are adding), we use a sequence of partial sums. That is, instead of considering the whole series all at once, we compute the sums sequentially and investigate the limit of the sequence of the partial sums.

a1, a2, a3, a4 , K, an, K

S1 = a1

S2 = a1 + a2

S3 = a1 + a2 + a3 = S2 + a3

M

Sn = a1 + a2 + a3 +K+ an = Sn−1 + an

Sn{ } We say that if the sequence of partial sums converges, then the series is convergent.

If the sequence of partial sums diverges, then the series diverges.

Arithmetic series: the sum of terms of an arithmetic sequence

Geometric series: the sum of terms of geometric sequence

Sn{ }




8.16. Sigma/Summation notations (section 12 – 5, pages 794 – 800)

We can also use shorthand for series in general. The shorthand requires the explicit definition of the sequence, and an index set (the set of positive integers, or a subset of the positive integers). Recall that when we discussed sequences, we said that formally, sequences are defined as functions on the positive integers or subsets of positive integers. The index set represents the domain of the sequence.

This shorthand for series is call the summation or sigma notation

( )( )

( )final index value end count

initial index value start countexplicit term of the sequence∑

1

k

nn

a=

∑

ending value of n

starting value of n The explicit definition of the terms of the sequence that are being added

ann=1

k

∑ = a1 + a2 + a3 +K+ ak for finite series

for infinite series

ann=1

∞

∑ = a1 + a2 + a3 +K+ ak−2 + ak−1 + ak + ak+1 + ak+2 +K

Example: Express the following series using sigma notation and find the sum (value of the series)

1+ 2 + 3+ 4 + 5 + 6 + 7 + 8 + 9 +10 Procedure

a) Determine the start point and end point.

Start point is 1 End point is 10

b) Find the explicit term for the underlying sequence

c)

d) sum = ????

an = n

nn=1

10

∑ =1+ 2 + 3+ 4 + 5 + 6 + 7 + 8 + 9 +10

Example: Express the following series using sigma notation and find the sum (value of the series)

(Gauss did this in his early years: You can investigate the story of Gauss with your class) 1+ 2 + 3+ 4 + 5 + 6 +K+100


Procedure

n

n=1

100

∑ =1+ 2 + 3+ 4 +K+100

1 + 2 + 3 + K + 99 100

100 + 99 + 98 + K + 2 1

101 + 101 + 101 + K + 101 101

2S =100 101( )= k k +1( )

S =12

k k +1( )

n

n=1

100

∑ =1+ 2 + 3+ 4 +K+100 = 12

k k +1( )=12

•100 • 101( )= 50 101( )= 5050.

Guided Practice: page 798, numbers 8 – 12, 27 - 41

Practice rewriting the following in a list (expanded form or progression)

Page 798, numbers 14 – 25.



8.17. Series Convergence (see section 12 – 4, pages 786 – 793)

a. Finite series are always convergent.

b. If the series is convergent, then ann=1

∞

∑ limn→∞

an = 0 . Conversely, if the or limn→∞

an ≠ 0 limn→∞

an does

not exist, then the series is divergent. ann=1

∞

∑Let’s explore c. Investigate arithmetic series

= ann=1

∞

∑ a1 + n −1( )dn=1

∞

∑ = a1 + d n −1n=1

∞

∑n=1

∞

∑ , shows that except the constant difference d =

0, an infinite arithmetic series will be divergent.

d. Graphically

Let f(x) = and f x( )= a1 + (x −1)d f n( )= an The graph of f(x) is linear, and hence the divergence

e. Investigate convergence of geometric series

= = S

Recall when we found the sum of the first 100 natural numbers, we added the series twice and then divided by two. We will want to do a similar thing with the geometric series.

ann=1

∞

∑ a1rn−1

n=1

∞

∑

rS − S = S r −1( )rS − S = a1r

n − a1 = a1 rn −1( )S r −1( )= a1 rn −1( )

S =a1 rn −1( )

r −1

1) If r <1, then , which means that limn→∞

rn = 0 S =a1

1− r

2) If r ≥1, then S is divergent

Graphically

Use f x( )= a xn( ) to explore the convergence of geometric series with different x values: a) x ≥1 b) x <1



Investigate Convergence of Series of reciprocals of factorials 1n!

∑

Explore Convergence of Series of reciprocals of powers 1n p∑

Explore with your students the ratio and comparison tests of section 12 - 4



8.18. Finite differences and pattern of power sequences

There are many more sequences that are neither arithmetic nor geometric, or Fibonacci.

Finite differences (difference equations) is a process that uses concepts of calculus to generate explicit definitions of sequences. Remember your work with slopes. When a function is linear (i.e., lines), you found that the slope is a constant. Or said another way, the rate of change is a constant. If a sequence is arithmetic, we have a constant difference d (this represents the rate of change). And we have shown that the explicit definition of the arithmetic sequence is linear.

If on the other hand, the rate of change of the rate of change is a constant (other than zero), then we have a quadratic situation. If the rate of change of that rate of change of the rate of change is a constant, then we have a cubic situation. And so on.

Once we determine the power of the sequence, we can use the graphing calculator curve fitting function, or matrices to determine the explicit definition of the sequence. Explicit definitions are very important, because they allow us to make quick predictions of any specific term of the sequence.

Explore some examples with your students.

Example: Handshake problem




8.19 Activities:

1. More Fibonacci (pages 813, page 40)

Investigate the following ratios of the terms of the Fibonacci sequence:

an =Fn +1

Fn

as n gets large

2. Figurate Numbers

Investigate the following sequences: Write both explicit and recursive definitions Make predictions Construct the following figurate patterns

(a) Triangular Numbers (b) Square Numbers (c) Pentagonal Numbers (d) Cubic Numbers

3. Folding/Creases/Planar Regions and Lines/Spread of Rumor

a) i) Fold a long strip of paper and count the creases ii) Organize the data in a table (number of folds by number of creases) iii) Organize the data in a table (number of folds by number of regions) iv) Use the data to predict number of creases and regions given number of folds

b) i) Divide the plane into regions using lines ii) Organize the data into a table (number of lines by maximum number of regions) iii) Use the data to predict the maximum number of regions if the number of lines is

specified

4. Zeno’s paradox

Suppose a student wants to walk across the room. To accomplish that, the student first goes half way, then half of the remaining distance, then half of the remaining distance, and so on. Will the student ever reach the other end of the room?

Why? Or why not?

Can you use what you have learned about sequences and series to solve this problem mathematically?

5. Zeno from Geometric Perspective (section 12 – 4, pages 786 – 790, page 792 number 34)

a) Suppose you start with an area of 2 square inches, then add half that area (that is 1 square inches), then add half the area you just added, and so on. What will be your combined area if you continue the process indefinitely?

b) Spiral growth/Sierpinski triangle (see section 12-7, page 782, number 45)



i) Sierpinski carpet: Remove the center one-ninth of a square of side 1. Now remove the centers of the eight smaller squares, then remove the centers of the 64 even smaller squares. What is the total area of all the removed squares?

ii) Sierpinski triangle: Remove the center fourth (an equilateral triangle) of an equilateral triangle, say with side length of 1 inch. Now remove the center fourths (also equilateral triangles) of the remaining three smaller equilateral. What is the total area of the removed triangles? Later we will use mathematical induction to show that remaining area after n equilateral triangles have been removed is

An =3n 34n +1 , for n ≥ 0, and the perimeter after n equilateral triangles have been

removed is Pn =3n+1

2n , for n ≥ 0.

6. Medication/decay/toxicity

Page 782, number 44

7. Reasoning and proof (mathematical induction: domino effect) (See section 12 – 9, pages 822 – 828)

Mathematical Induction allows us to proof assertions about the natural numbers. Since sequences are assertions about natural numbers, mathematical induction will allow us to determine if our claims are true or not. This tool works on sequences on natural numbers or sequences of partial sums (i.e., series).

Suppose you have an assertion such as the sum of the first n positive integers is n n +1( )

2. That is

1+ 2 + 3+ 4 +K+ n =

n n +1( )2

.

We check the initial step, in this case, when n = 1.

Then we hypothesize either that (a) the assertion is true for some value of n, say k. Then show that the assertion is true for n = k+1. Or (b) we can hypothesize that the assertion is true for all n from 2 to k. Then show the assertion is true for k+1.

The idea is that if it is true for k+1, it is also true for k+2, and so on. Hence the assertion must be true for all values of n. In other words, if you can prove the initial case, hypothesize about an arbitrary case, and show that the assertion is true for the next case, then you can show that the assertion continues to be true for subsequent cases. Hence the domino effect.

See section 12-9, pages 822 – 828.

8. Section 12–7 has investigations of other useful series that you should explore with your students.



9. Capstone

a) Tower of Hanoi (provide actual models for students to use or use available internet applets)

Ask students to generate the fewest number of steps needed to resolve up to 64 disks (the original problem for the monks). Hopefully, they will have used the ideas in this section/unit to look at fewer cases and come up with a pattern from which they can predict the 64th case.

Suppose it takes about one second to move a disk, how long will it take to complete a ten-disk problem? What about the 64-disk problem?

b) Assign groups of students to investigate national brand electronic stores’ growth (# of stores) from 1992 to 2000 (period of expansion in the economy), and look at other periods 2000 to 2004, 1980 to 1992, etc. Sequence the data annually. Tie this activity into piecewise, power, exponential, or logistic functions. You can have the students produce a poster and write a report on their analyses. What conclusions can they draw from their analyses?



8.20 Additional Resources

Core-Plus Mathematics Project. (2003). Contemporary Mathematics in Context, Course 3, Unit 7. Columbus, OH: Glencoe/McGraw-Hill

Core-Plus Mathematics Project. (2003). Contemporary Mathematics in Context, Course 4, Unit 4. Columbus, OH: Glencoe/McGraw-Hill

Sobel, M. A. (Ed.) (1988). Enrichment in secondary school mathematics. Reston, VA: National Council of Teachers of Mathematics.

Stewart, J. Calculus (5th ed.). USA: Thomson - Brooks/Cole.

NCTM Illuminations

MathForum


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