A flawed predictionAuthor(s): John GeorgeSource: The Mathematics Teacher, Vol. 95, No. 4 (April 2002), pp. 308, 310Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/20871020 .

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(Continued from page 296)\

one side of one of the internal polygons.

Fig. 2 (Roof) A regular heptagon

Brendan Roof Jack Roof jchemroof@sprintmail. com Poulsbo,WA 98370

Sums of powers Numerous methods have been developed to find the sums of powers of numbers 1* + 2k + +

nk,k = 1, 2,... A few of them are suitable for high school mathe matics teaching, including the older one presented by S. M. Elzaidi in "Reader Reflections: Sums of powers of numbers," on pages 340 and 399 of the May 2001 issue of the Mathematics Teacher. I would like to give a

simple method that is based on an interesting property, as follows.

Observe that

(5?4?3)-(4-3-2) 4.3 = v-LA-1

3

(8-7-6*5)-(7-6?5-4) 7-6-5 =

^-}-, 4

v ln + 2)(n + l)n-(n + i)n(n-i) (n

+ l)n

= X--tl-U-LS

and so on. Thus, in general, if we let

M = (n+j-l)(n+j-2) (n + 2)(n + \)n,

L = (n+j)(n+j-D*

(n + 2)(n + l)n, and

R = (n+j- l)(n +j-2) (n + l)n(n - 1),

where j is any positive integer, then

7 + 1 that is,

(1) (n+j-l)(n +j-2). in + 2)(n +

{n + j){n + j-l)?- -(w

+ 2)(7i

+ l)n "

7+1

(n + j-l)(n

+ j-2y- -(71

+ 1)1(71-1)

7+1

In the preceding expression, if j is a fixed number and if the first term on the right-hand side is a function of n, say, fin), then the second term is exactly fin

- 1). The following examples show

that by effectively applying equa tion (1), the calculation of sums of powers of numbers and other types of series can be made simple.

Example 1

Find the sum Sn = 1 + 2 + + n. Here S, = 1 and Sn

- SnA = n,

which corresponds to j = 1 in equation (1).

Sr ~

Sn-i - n

= ?(n

+ l)n-?n(n-l\

So Sn must be equal to (1/2) in + l)n + C, where C is a con stant, or [Sn

- (U2)(n + Dti) is a sequence of constants whose first term is Sx

- (1/2)(2)(1) = 0. Therefore,

Sn=^i(n + l).

Example 2

FindSn = l2 + 22 + .-. + 7i2. In this case, Si = 1. By applying equation (1) to Sn

- Sn_h we get

Sn ~

^7i-i = r2

We see that

S? --(n + 2)(n + l)n + -(n + l)n

is a sequence of constants whose first term is

Hence,

Sn =-(/i +

2^ +

i)ai--(w +

i)w 3 2

= ^(fl

+ l)(27l + l).

Example 3

Find the sum of l3 + 23 + + ns. Here we can replace ns with

{ns - n) + n = (n + l)n(zi - 1) + n

and then apply result (1), as follows:

Sn - =

(n + 1)71(71 -1)+7i

= ^(w

+ 2)(w + l)/i(/i-l)

-J(, + l),(,-l)(,-2)

+- (n +1 )ti --n(n -l).

Thus, the sequence

is one having constant terms. We note that S\ = 1, and the first term of this sequence is

So

Sn =i(7i + 2)(7i

+ l)n(?-l)

+ ^(7i

+ l)w

Example 4

FindSn = l- 2 + 2- 3 + .-.+ n(n + 1).

Since S\ = 2 and since

= -(w + 2)(n + l)/i

we know that

s- =|(B+2XB+1)n+si-|(3X2X1) = ^(n+l)(n

+ 2). Sidney H. Kung shkung@tu.infi.net University of North Florida Jacksonville, FL 32224

Side and yeliow make rsreen

Graphing compound linear inequalities in one variable can be

made more accessible to students if they use blue and yellow trans parent plastic arrows, similar to those shown in figure 1 (Metz), that are made from report covers. The pieces with the curved ends are used when a strict inequality, such as x> -1, is represented.

When graphing the compound inequality x> -1 and x < 1, we use two number lines. On the first, a piece of blue plastic is used to show x>-l and a yellow piece is used to show x < 1. See figure 2 (Metz). On the second graph, we can use a pencil to shade the area of the graph that is green, that is, the intersection of the two sets. In general, we show one inequality with blue and one with yellow and then use the second number line to shade the part of the number line that is colored green. If "or" is used in

place of "and," we shade all num bers that are covered with plastic.

-5-4-3-2-10 1 2 3 4 5

H?I?h-H??l?l?l?H

-5 -4 -3 -2 -1 0 1 2 3 4 5

Fig. 2 (Metz) x> -1 and x < 1

has solution (-1, 1].

Each student must have the set of four pieces; this demon stration is not meant to be a classroom demonstration conduct

ed by the teacher but is rather an

activity to engage each student. James Metz metz@hawaii.edu

Diane Komenaka dkomenak@hawaii. edu

Kapiolani Community College Honolulu, HI 96816

A flawed prediction As a mathematics and economics

(Continued on page 310)

308 MATHEMATICS TEACHER

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student, I was enthusiastic about the "Media Clips" that discussed patents on page 383 in the Octo ber 2000 issue of the Mathematics Teacher. After solving the prob lem, I checked my solutions. My answers corresponded to the answers in the journal, but on further investigation, I found a flaw in the answer.

The flaw was in the regression that was used to model the data for the growth of patents over time. According to the calculator, the quadratic regression equation was the best-fitting model, but this model caused an error in predicting future growth. Data were given for increments of one million patents and the years cor

responding to those numbers. After the quadratic equation that models the data was calculated, the equation was used to predict the year that the next increment of one million patents would be reached. This prediction is flawed. The prediction states that fifteen years will be needed for another million patents to ac cumulate. The previous million only took eight years to accumu late, and the million before that took fifteen years. This behav ioral pattern is not the one for a

quadratic model. If the data are

parabolic, then the time needed to reach each future increment of one million should be less than the previous amount of time neces sary because y

= x.

The regression that best fits the data may be quadratic, but the quadratic regression is not necessarily the right model for the data. Through graphing the regression equation over the data?or more accurately, by graphing the regression's residuals?we can see that the rate of growth is underestimated. This underestimation may appear to be insignificant, but as x is squared, the error becomes magnified over time.

In regard to the problem itself, more data points should have been considered when trying to model the data. The time span for the data is from 1836, and the data points only show when an increment of one million has been

reached. A complete table of data for the number of patents filed each year can be found at the U.S. Patent and Trademark Office Web site (www.uspto.gov/web/offices /ac/ido/oeip/tafi^issuyear.htm). When dealing with such data

as patents, which vary according to the economy, it is conceivable that no true model exists for the data. The data that are being modeled were present for eco nomic prosperity, as well as for depression.

Believing that the condition of the economy would not affect the number of patents registered in a

given year is ignorant. When modeling data, consistency is the key to having a good fit. The data depend on too many variables that cannot be held constant. The stock market is another example.

With a fluctuating economy, vari ables change too often and too randomly to be modeled by an

equation. To answer the question, "What

equation best models the data?" the correct answer is no equation, because the real-world data depend on too many variables.

John George georgj3@rpi.edu St. John's High School Shrewsbury, MA 01545

George makes a valid point that the equation that minimizes the sum of the squares of the residuals of the given data points is not necessarily the equation that gives the most realistic predictions. In the patents example, if the point corresponding to the year 1836 is omitted and if exponential regression is used, the predicted year for 7 million patents is 2006. This prediction is more realistic than the one given by the quad ratic regression.?Ed.

Correction Two references were inadvertent ly omitted from "A Graphical

Approach to Understanding the Fundamental Theorem of Algebra," by Sudhir Kumar Goel and Denise T. Reid, on pages 749-59 of the December 2001 issue of the

Mathematics Teacher. The ref erences are as follows:

"The Fundamental Theorem of Algebra." SOS Mathematics. www.sosmath.com/algebra/factor /fac04/fac04.html. World Wide

Web.

"The Fundamental Theorem of

Algebra." www-groups.dcs.st

-Andrews.ac.uk/history

/HistTopics. Select Fund _theorem_of_algebra.html. World Wide Web. 1996. ?Ed.

Problem 20, December 2001 An error or misprint exists in the solution to problem 20 in the December 2001 "Calendar" in the Mathematics Teacher. The prob lem reads as follows:

If a and b are integers, each

greater than 1, and iaiaia = b, determine the smallest possible value of a + b.

The sixth line from the end of the solution says, "That is, either a = 28and& = 37 or a = 38 and 6 = 27."

I think that it should have read, "That is, either a = 28 and b = 21 ora = 38 and& = 37."

The answer to the original question, the smallest possible value of a + b, is then 384.

Charles K. Walsh St. Mary's County Public

Schools Leonardtown, MD 20650

The correct solution should read as follows:

384. Squaring both sides of the given equation shows that b2 =

aVaVa, so

Finally, we see that

?6

8 (b*1 3 a =

a,

so bs = a1. For a and b to both be integers, a must be an

eighth power and b must be a seventh power, and both must be greater than 1. For a + b to be as small as possible, a and 6 should be powers of 2 and 3. That is, either a = 28 and b = 27 or a = 38 and 6 = 37. The choice that results in the smallest sum is the former; that is, 256 + 128 = 384, as compared to 6561 + 2187 = 8748, soa = 28 = 256 and& = 27 = 384.

We also thank Chris Fisher, of Miamisburg, Ohio, for calling the error to our attention.?Ed.

In other IUCTM journals Readers of the Mathematics Teacher might enjoy the April focus issue on "The Magic of Number" in Mathematics Teaching in the Middle School. Articles in that issue include the following:

"X"-tending The Fibonacci Sequence," by Glenn Moran

"Investigating Limits in Num ber Patterns," by Darin Beigie

And from the April issue of Teaching Children Mathematics:

"Developing Spatial Under standing through Building Polyhedrons," by Rebecca C. Ambrose and Karen Falkner

For a complete listing of the con tents of all the journals, see the NCTM Web site at www.nctm.org.

Share Your Journal Sharing articles from the Mathematics Teacher

is an easy way to gain support for trying new activi ties or techniques in your mathematics classroom.

Sharing and talking about the journal yield great returns for you and your fellow teachers. Try it, and

then tell us about your experiences by writing to "Reader Reflections," NCTM,

1906 Association Drive, Reston, VA 20191-9988.

310 MATHEMATICS TEACHER

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