# I remember that when I

Size: px
Start display at page:

Transcription

1 8. Airthmetic and Geometric Sequences ARITHMETIC AND GEOMETRIC SEQUENCES Whenever you tell me that mathematics is just a human invention like the game of chess I would like to believe you. But I keep returning to the same problem. Why does the mathematics we have discovered in the past so often turn out to describe the workings of the Universe? John Barrow Two kinds of regular sequences occur so often that they have specific names, arithmetic and geometric sequences. We treat them together because some obvi- ous parallels between these kinds of sequences lead to similar formulas. This also makes it easier to learn and work with the formulas. The greatest value in this association is understanding how the ideas are related and how to derive the formulas from fundamental concepts. Anyone learning the formulas this way can recover them whenever needed. Both arithmetic and geometric sequences begin with an arbitrary first term, and the sequences are generated by regularly adding the same number (the common difference in an arithmetic sequence) or multiplying by the same number (the common ratio in a geometric sequence). Definitions emphasize the parallel features, which examples will clarify. I remember that when I was about twelve I learned from [my uncle] that by the distributive law times equals. I thought that was great. Peter Lax Definition: arithmetic and geometric sequences Arithmetic Sequence a a and a n a n d for n The sequence a n is an arithmetic sequence with first term a and common difference d. Geometric Sequence a a and a n r a n for n The sequence a n is a geometric sequence with first term a and common ratio r. The definitions imply convenient formulas for the nth term of both kinds of sequences. For an arithmetic sequence we get the nth term by adding d to the first term n times; for a geometric sequence, we multiply the first term by r, n times. Formulas for the nth terms of arithmetic and geometric sequences For an arithmetic sequence, a formula for the nth term of the sequence is a n a n d. () For a geometric sequence, a formula for the nth term of the sequence is a n a r n. () The definitions allow us to recognize both arithmetic and geometric sequences. In an arithmetic sequence the difference between successive terms, a n a n,is always the same, the constant d; in a geometric sequence the ratio of successive terms, a n, is always the same. a n

2 45 Chapter 8 Discrete Mathematics: Functions on the Set of Natural Numbers EXAMPLE Arithmetic or geometric? Thefirstthreetermsofasequence are given. Determine if the sequence could be arithmetic or geometric. If it is an arithmetic sequence, find d; for a geometric sequence, find r. Strategy: Calculate the differences and /or ratios of successive terms. (a),4,8,... (b) ln,ln4,ln8,... (c),, 4,... (a) a a 4, and a a Since the differences are not the same, the sequence cannot be arithmetic. Checking ratios, a 4 a, and a 8, so the sequence could be geometric, with a common ratio a 4 r. Without a formula for the general term, we cannot say anything more about the sequence. (b) a a ln 4 ln ln 4 ln, and a a ln 8 ln 4 ln 8 4 ln, so the sequence could be arithmetic, with ln as the common difference. As in part (a), we cannot say more because no general term is given. (c) a a,anda 6 a 4. The differences are a not the same, so the sequence is not arithmetic. a, and a 4 a, so the sequence is not geometric. Note that the sequence in part 4 (a) could be geometric and the sequence in part (b) could be arithmetic, but in part (c) you can conclude unequivocally that the sequence cannot be either arithmetic or geometric. EXAMPLE Arithmetic or geometric? Determine whether the sequence is arithmetic, geometric, or neither. (a).6n (b) n (c) a n ln n (a) a a , and a a From the first three terms, this could be an arithmetic sequence with d.6. Check the difference a n a n. a n a n.6 n.6n.6. The sequence is arithmetic, with d.6. (b) a a 4, and a a 8 4 4, so the sequence is not arithmetic. Using the formula for the general term, a n a n n n. The sequence n is geometric, with as the common ratio.

3 8. Airthmetic and Geometric Sequences 45 (c) a n a n ln n ln n ln n. The difference depends on n, so n the sequence is not arithmetic. Checking ratios, a n ln n,sotheratio a n ln n also changes with n. The sequence is neither arithmetic nor geometric. EXAMPLE Arithmetic sequences Show that the sequence is arithmetic; find the common difference and the twentieth term. (a) a n n (b) 50,45,40,...,55 5n,... (a) The first few terms of a n are,,5,7,...,fromwhich it is apparent that each term is more than the preceding term; this is an arithmetic sequence with first term and common difference a andd. Check to see that a n a n. To find a 0, use either the defining formula for the sequence or Equation () for the nth term: a or a 0 a 9d 9 9. (b) If b n 55 5n, then b n b n 55 5 n 55 5n 5. This is an arithmetic sequence with a 50, d 5, and so b Given the structure of arithmetic and geometric sequences, any two terms completely determine the sequence. Using Equation () or (), two terms of the sequence give us a pair of equations from which we can find the first term and either the common difference or common ratio, as illustrated in the next example. EXAMPLE 4 Arithmetic sequences Suppose a n is an arithmetic sequence with a 8 6anda 4. Find a, d, and the three terms between a 8 and a. From Equation (), a 8 a 7d, anda a d, from which the difference is given by a a 8 4d. Use the given values for a 8 and a to get 4 6 4d, or d 5. Substitute 5 47 for d in 6 a 7d and solve for a, a.findthe three terms between a 8 and a by successively adding 5 : a 9 a 8 5 7, a 0 a 9 5, a a 0 5. Therefore, a 9 is 7, a 0 is, and a is. EXAMPLE 5 Geometric sequences Determine whether the sequence is geometric. If it is geometric, then find the common ratio and the terms a, a, and a 0. (a) n (b),, 9,..., n,...

4 454 Chapter 8 Discrete Mathematics: Functions on the Set of Natural Numbers Strategy: The property that identifies a geometric sequence is the common ratio: the values a, a, a 4,... a a a must all be the same. For a geometric sequence, use Equation (). (a) Thefirstfewtermsare,4,8,6,...,each of which is twice the preceding term. This is a geometric sequence with first term a, and common ratio given by r a n a n n. Using a n n n, (b) Consider the ratio a a 8 and a a n a n n n, so the sequence is geometric with a andr. Using a n n, we get a, a ar 9,anda 0 ar Partial Sums of Arithmetic Sequences There is a charming story told about Carl Freidrich Gauss, one of the greatest mathematicians of all time. Early in Gauss school career, the schoolmaster assigned the class the task of summing the first hundred positive integers, That should have occupied a good portion of the morning, but while other class members busied themselves at their slates calculating, 6, 6 4 0, and so on, Gauss sat quietly for a few moments, wrote a single number on his slate, and presented it to the teacher. Young Gauss observed that and 00 add up to 0, as do the pair and 99, and 98, and so on up to 50 and 5. There are fifty such pairs, each with a sum of 0, for a total of , the number he wrote on his slate. This approach works for the partial sum of any arithmetic sequence, and we will use the method to derive some useful formulas. However, the ideas are more valuable than memorizing formulas. If you understand the idea, you can recreate the formula when needed. To find a formula for the nth partial sum of an arithmetic sequence, that is, the sum of n consecutive terms, pair the first and last terms, the second and next-to-last, andsoon;each pair has the same sum. Infact,itiseasiertopairalltermstwice, as illustrated with Gauss sum: S S S The sum on the right has 00 terms, so S Dividing by, S For the general case, pairing the terms in S n and adding gives S n n a a n because there are n pairs, each with the same sum. Dividing by yields the desired formula.

5 8. Airthmetic and Geometric Sequences 455 Partial sums of an arithmetic sequence Suppose a n is an arithmetic sequence. The sum S n of the first n terms is given by S n n a a n () The formula is probably most easily remembered as n times the average of the first and last terms. EXAMPLE 6 Partial sums For the sequence a n n, (a) evaluate the sum S 5 5 k k and (b) find a formula for S n. Strategy: Let a Follow the strategy. n n. To find S 5 from Equation () requires a and a 5, (a) By Equation (), S 5 5 a a 5.Now,finda and a 5. which the formula for a n can provide. For (b), substitute for a a and a and n fora n in Equation () and simplify Thus, S (b) In general, S n n a a n Hence, S n n. n n n n n. EXAMPLE 7 Arithmetic sequence Thesumofthefirsteighttermsofan arithmetic sequence a n is 4; the sixth term is 0. Find a formula for a n. For a n, first find a and d. Since a 6 a 5d, a 5d 0. Express S 8 in terms of a and d, 8 a a 7d S 8 4 a 7d. Since we are given S 8 4, Equation () states that 4 a 7d 4. This gives a pair of equations to solve for a and d. a 5d 0 a 7d 6 We find d anda 0. Therefore, the nth term is a n a n d 0 n n. Partial Sums of Geometric Sequences The idea of pairing terms, which works so well for arithmetic sequences, does not help with a geometric sequence. Another idea does make the sum easy to calculate

6 456 Chapter 8 Discrete Mathematics: Functions on the Set of Natural Numbers though. Multiply both sides by r and subtract: S n a ar ar ar n rs n ar ar ar n ar n S n rs n a ar n Thus, S n r a r n. If r, dividing both sides by r yields a formula for S n. Partial sums of a geometric sequence Suppose a n is a geometric sequence with r. The sum of the first n terms is S n a r n r (4) In the special case where r, the geometric sequence is also an arithmetic sequence, and S n a a a a na. Strategy: Since it is given that the sequence is a geometric, find the common ratio r a andthenuse a Equations () and (4). EXAMPLE 8 Partial sum Find a n and S n for the geometric sequence,, 6 Follow the strategy. We know that a and a is 6. The common ratio is r a 6 a n ar n. From Equation (), n n. Since r, r and r n n. Applying Equation (4) gives S n a r n r Therefore, n a n n and S n n n. EXAMPLE 9 Limit of a sum (a) Find the sum of the first 5, 0, and 00 terms of the geometric sequence from Example 8. (b) Draw a graph of S n in 0, c 0,, where c is the number of pixel columns of your n calculator (see inside front cover). Trace to find the smallest integer n for which the y-value is displayed as.

7 8. Airthmetic and Geometric Sequences 457 [0, c] by [0, ] FIGURE y (/)( / x) (a) InExample8wefoundaformulaforthenth partial sum, S n. Substituting 5, 0, and 00 for n, n S , S S The term has 0 zeros immediately following the decimal point. That 00 means that S 00 is so near that a calculator cannot display the difference except as. (b) In the window 0, c 0, we see a graph something like Figure. Because calculators display trace coordinates differently, you may see something other than ours, but somewhere between 5 and 5, you should see the y-value displayed something like , the nearest your calculator can come to displaying. Looking Ahead to Calculus: Infinite Series As indicated above, each sequence a n is associated with a sequence of partial sums S n, where S n a a a n. What happens to S n as n gets larger and larger, that is, as we add more and more terms? We are considering an infinite sum written as a a a,orinsummation notation, a n. n This is called an infinite series. Since we cannot add an infinite set of numbers, we need instead the notion of a limit. In one sense, calculus is the study of limits. It is beyond the scope of this book to deal with infinite series in general, but for a geometric sequence a n,we can at least get an intuitive feeling for what happens to S n as n becomes large. In Examples 8 and 9, where a n and S n n ( n), it is reasonable to assume that gets close to 0 as n becomes large. In calculus notation n lim na n 0, from which lim S n na. We say that the infinite series, n n n 6. n converges to, and we write In general, we associate each geometric sequence ar n with an infinite geometric series ar n a ar ar ar n. n

8 458 Chapter 8 Discrete Mathematics: Functions on the Set of Natural Numbers The only meaning we give to this infinite sum is the limit of the sequence of partial sums, lim S n lim na na a r n, r which depends on lim na r n. Looking at different values of r, we conclude that if r is any number between and, then lim na r n 0, from which a r n lim S n lim na na r a r. Infinite geometric series Associated with every geometric sequence ar n is an infinite geometric series ar n a ar ar ar n. n If r, then the series converges to a, and we write r ar n a ar ar ar n n a r. (5) If r, then the infinite series does not have a sum, and it diverges. Repeating decimals. In Section. we said that the decimal representation of any rational number is a repeating decimal. The following example illustrates how we can use an infinite geometric series to express a repeating decimal as a fraction of integers. EXAMPLE 0 Repeating decimal Write in terms of an infinite geometric series, then use Equation (5) to express.45 in the form p, where p and q are integers. q The terms following 45 0 form an infinite geometric series with a and 000 r Since r is between and, we may use Equation (5) to express the sum as Therefore, 7 0 and.45 represent the same number. Functions represented by infinite series. The infinite series x x is geometric (with a andr x), so if x is any number between and, the

9 8. Airthmetic and Geometric Sequences 459 series converges: x x x x. Hence the function f x, where x, can be represented by the x infinite series x x. An important topic arises in calculus when we represent functions by infinite series. For instance, it can be shown that the function F x sin x is also given by sin x x x! x 5 5! x 7 7!. The representation for sin x is not a geometric series, but it does converge for every real number x. Itfollowsthatsinxcan be approximated by polynomial functions consisting of the first few terms of the infinite series. For example, if we let p x be the sum of the first four terms, p x x x! x 5 5! x 7, then p x sin x. 7! Evaluating at x 0.5, sin 0.5 p To see how good this approximation is, use your calculator to evaluate sin 0.5 (in radian mode). In fact, your calculator is probably designed to use polynomial approximations to evaluate most of its built-in functions. Following are series representations for some important functions we have studied in Chapters 4 and 5. sin x x x! x 5 5! x 7 cosx x 7!! x 4 4! x 6 6! e x x x! x! e x x x! x! EXERCISES 8. Check Your Understanding Exercises 6 True or False. Give reasons.. If a n is an arithmetic sequence, then a 6 a a 8 a 5.. The sequence beginning,,, 4 6 8,...could be an arithmetic sequence.. If c n is a geometric sequence, then c 5 r. c 4. The sequences a n and b n given by a n n and b n log 00 n are identical. 5. In a geometric sequence if the common ratio is negative, then after a certain point in the sequence, all the termswillbenegative. 6. In an arithmetic sequence if the common difference is negative, then after a certain point in the sequence, all the terms will be negative. Exercises 7 0 Fill in the blank so that the resulting statement is true k. k k. k

10 460 Chapter 8 Discrete Mathematics: Functions on the Set of Natural Numbers Develop Mastery Exercises 0 Arithmetic Sequences The first three terms of an arithmetic sequence are given. Find (a) the common difference, (b) the sixth and tenth terms, and (c) the sum of the first ten terms..,6,9,.... 8,, 4,.... 4,, 6,... 4.,, 5, , 5, , 0., 0.40,... 7.,,, ,, 5, ln,ln4,ln8, ln e, 4,7,... Exercises 0 Geometric Sequences The first three terms of a geometric sequence are given. Find (a) the common ratio, (b) the sixth and eighth terms, and (c) the sum of the first five terms..,,,... 9.,,,.... 8,6,,... 4., 0.5, 0.5,... 5.,,,... 6., 7.,.5, 0.75, ,,, ,,,... 0.,,,... Exercises 8 Arithmetic or Geometric? The first three terms of a sequence are given. Determine whether the sequence could be arithmetic, geometric, or neither. If arithmetic, find the common difference; if geometric, give the common ratio..,, 4,...., 4, 8,....,,, ln, ln, ln,... 6.,4,9, , 0.00, , e, e, e,... Exercises 9 6 Arithmetic Sequences Assume that the given information refers to an arithmetic sequence. Find the indicated quantities. 9. a 5, a 6 0; d, a 0. a 5, d ; a, a 0. a, a 8 5; d, S 8. a 8, a 9 ; S 4, S 6. a 8 5, S 8 64; a, S 4 4. a 6, S 6 8; a, S 6 5. a 5, d, a 4, a 6, S 6 6. a 5, a 8 4 ; a, a, S Exercises 7 44 Geometric Sequences Assume that the given information refers to a geometric sequence. Determine the indicated quantities. 7. a 4, a 6; r, a 6 8. a, a ; a 4, a 7 9. a 5 4, r ; a, S a 4 6, a 7 48; r, a 0 4. a 8, S 4; a 5, S 5 4. a 4, a 7 ; r, S a 8,a 5 0 ; a 80, S a, S ; a 6, S 6 Exercises Find x Determine the value(s) of x for which the given expressions will form the first three terms of the indicated type of sequence. 45., x, x ; arithmetic 46. x, x, x 6; arithmetic 47., x, x ; geometric 48. x, x, x 6; geometric 49. x, x, x ; arithmetic 50., x, x 4 ; geometric Exercises 5 56 Arithmetic or Geometric? Three expressions are given. Determine whether, for every real number x, they are the first three terms of an arithmetic sequence or a geometric sequence. 5. x, x, x 5 5. x, x,4 x 5. x, x, x 54. x, x, x 55. x, x, x 56. x, x, 4 x Exercises Infinite Series For the infinite series, (a) write out the first four terms, find the common ratio and aformulafors n.(b) Find the sum of the series; that is, find the limit of S n asngetslarge n n n n 4 5 n 59. n 60. n 4 n n Exercises 6 6 Sum of an Infinite Series Find the sum of the infinite geometric series Exercises6 64 Geometric Sequence, Partial Sums, Convergence The first three terms of a geometric sequence are given. (a) Use Equation (4) to find a formula for S n as a function of n, and draw a graph of S n. (b) Using Equation (5)findthelimitLof S n.(c) Trace to find the smallest valueofnforwhich S n L is less than 0.00, See Example ,.4,.44, ,., 0.96,...

11 8.4 Patterns, Guesses, and Formulas 46 Exercises Express as a quotient of two integers in reduced form. 65. (a).4, (b) (a).45, (b) (a).5, (b) (a) 0.7, (b) Evaluate the sum (a) How many integers between 00 and 000 are divisible by? (b) What is their sum? 7. Find the sum of all odd positive integers less than Find the sum of all positive integers between 400 and 500 that are divisible by. 7. If a b c and a, b, andcare the first three terms of a geometric sequence, show that the numbers log a 4, log b 4,and are three consecutive terms log c 4 of an arithmetic sequence. (Hint: Use the change of the base formula from page 8.) 74. In a geometric sequence a n of positive terms, a a and a 5 a 4 4. Find the first five terms of the sequence. 75. If the sum of the first 60 odd positive integers is subtractedfromthesumofthefirst60evenpositiveintegers, what is the result? 76. The measures of the four interior angles of a quadrilateral form four terms of an arithmetic sequence. If the smallest angle is 7, what is the largest angle? 77. In a right triangle with legs a, b, and hypotenuse c, suppose that a, b, andcare three consecutive terms of a geometric sequence. Find the common ratio r. 78. Theseatsinatheaterarearrangedinrowswith40 seats in the first row, 4 in the second, 44 in the third andsoon. (a) Howmanyseatsareinthethirty-firstrow?Inthe middle row? (b) How many seats are in the theater? 79. A rubber ball is dropped from the top of the Washington Monument, which is 70 meters high. Suppose each time it hits the ground it rebounds of the distance of the preceding fall. (a) What total distance does the ball travel up to the instantwhenithitsthegroundforthethirdtime? (b) What total distance does it travel before it essentially comes to rest? 80. Suppose we wish to create a vacuum in a tank that contains 000 cubic feet of air. Each stroke of the vacuumpumpremoveshalfoftheairthatremainsinthe tank. (a) How much air remains in the tank after the fourth stroke? (b) How much air was removed during the fourth stroke? (c) How many strokes of the pump are required to remove at least 99 percent of the air? 8. From a helicopter hovering at 6400 feet above ground level an object is dropped. The distance s it falls in t seconds after being dropped is given by the formula s f t 6t. (a) How far does the object fall during the first second? (b) Let a n denote the distance that the object falls during the nth second, that is, a n f n f n. Find a formula for a n. What kind of sequence is a n? (c) Evaluate the sum a a a, and then find s when t. Compare these two numbers. (d) Clearly this is a finite sequence since the object cannot fall more than 6400 feet. How many terms are there in the sequence? What is the sum of these terms? 8.4 PATTERNS, GUESSES, AND FORMULAS What humans do with the language of mathematics is to describe patterns. Mathematics is an exploratory science that seeks to understand every kind of pattern patterns that occur in nature, patterns invented by the human mind, and even patterns created by other patterns. Lynn Arthur Steen Arithmetic and geometric sequences are highly structured, and it is precisely because we can analyze the regularity of their patterns that we can do so much with them. The formulas developed in the preceding section are examples of what can be done when patterns are recognized and used appropriately.

### 10.2 Series and Convergence

10.2 Series and Convergence Write sums using sigma notation Find the partial sums of series and determine convergence or divergence of infinite series Find the N th partial sums of geometric series and

### 9.2 Summation Notation

9. Summation Notation 66 9. Summation Notation In the previous section, we introduced sequences and now we shall present notation and theorems concerning the sum of terms of a sequence. We begin with a

### Arithmetic Progression

Worksheet 3.6 Arithmetic and Geometric Progressions Section 1 Arithmetic Progression An arithmetic progression is a list of numbers where the difference between successive numbers is constant. The terms

### GEOMETRIC SEQUENCES AND SERIES

4.4 Geometric Sequences and Series (4 7) 757 of a novel and every day thereafter increase their daily reading by two pages. If his students follow this suggestion, then how many pages will they read during

### Lies My Calculator and Computer Told Me

Lies My Calculator and Computer Told Me 2 LIES MY CALCULATOR AND COMPUTER TOLD ME Lies My Calculator and Computer Told Me See Section.4 for a discussion of graphing calculators and computers with graphing

### 4/1/2017. PS. Sequences and Series FROM 9.2 AND 9.3 IN THE BOOK AS WELL AS FROM OTHER SOURCES. TODAY IS NATIONAL MANATEE APPRECIATION DAY

PS. Sequences and Series FROM 9.2 AND 9.3 IN THE BOOK AS WELL AS FROM OTHER SOURCES. TODAY IS NATIONAL MANATEE APPRECIATION DAY 1 Oh the things you should learn How to recognize and write arithmetic sequences

### Properties of sequences Since a sequence is a special kind of function it has analogous properties to functions:

Sequences and Series A sequence is a special kind of function whose domain is N - the set of natural numbers. The range of a sequence is the collection of terms that make up the sequence. Just as the word

### Section 1.3 P 1 = 1 2. = 1 4 2 8. P n = 1 P 3 = Continuing in this fashion, it should seem reasonable that, for any n = 1, 2, 3,..., = 1 2 4.

Difference Equations to Differential Equations Section. The Sum of a Sequence This section considers the problem of adding together the terms of a sequence. Of course, this is a problem only if more than

### IB Maths SL Sequence and Series Practice Problems Mr. W Name

IB Maths SL Sequence and Series Practice Problems Mr. W Name Remember to show all necessary reasoning! Separate paper is probably best. 3b 3d is optional! 1. In an arithmetic sequence, u 1 = and u 3 =

### Chapter 31 out of 37 from Discrete Mathematics for Neophytes: Number Theory, Probability, Algorithms, and Other Stuff by J. M.

31 Geometric Series Motivation (I hope) Geometric series are a basic artifact of algebra that everyone should know. 1 I am teaching them here because they come up remarkably often with Markov chains. The

9.3 Solving Quadratic Equations by Using the Quadratic Formula 9.3 OBJECTIVES 1. Solve a quadratic equation by using the quadratic formula 2. Determine the nature of the solutions of a quadratic equation

### SOLVING EQUATIONS WITH RADICALS AND EXPONENTS 9.5. section ( 3 5 3 2 )( 3 25 3 10 3 4 ). The Odd-Root Property

498 (9 3) Chapter 9 Radicals and Rational Exponents Replace the question mark by an expression that makes the equation correct. Equations involving variables are to be identities. 75. 6 76. 3?? 1 77. 1

### APPLICATIONS AND MODELING WITH QUADRATIC EQUATIONS

APPLICATIONS AND MODELING WITH QUADRATIC EQUATIONS Now that we are starting to feel comfortable with the factoring process, the question becomes what do we use factoring to do? There are a variety of classic

### Quick Reference ebook

This file is distributed FREE OF CHARGE by the publisher Quick Reference Handbooks and the author. Quick Reference ebook Click on Contents or Index in the left panel to locate a topic. The math facts listed

### Unit 1 Number Sense. In this unit, students will study repeating decimals, percents, fractions, decimals, and proportions.

Unit 1 Number Sense In this unit, students will study repeating decimals, percents, fractions, decimals, and proportions. BLM Three Types of Percent Problems (p L-34) is a summary BLM for the material

PYTHAGOREAN TRIPLES KEITH CONRAD 1. Introduction A Pythagorean triple is a triple of positive integers (a, b, c) where a + b = c. Examples include (3, 4, 5), (5, 1, 13), and (8, 15, 17). Below is an ancient

### 1.2. Successive Differences

1. An Application of Inductive Reasoning: Number Patterns In the previous section we introduced inductive reasoning, and we showed how it can be applied in predicting what comes next in a list of numbers

### A Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions

A Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions Marcel B. Finan Arkansas Tech University c All Rights Reserved First Draft February 8, 2006 1 Contents 25

### Math Review. for the Quantitative Reasoning Measure of the GRE revised General Test

Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important

### A.2. Exponents and Radicals. Integer Exponents. What you should learn. Exponential Notation. Why you should learn it. Properties of Exponents

Appendix A. Exponents and Radicals A11 A. Exponents and Radicals What you should learn Use properties of exponents. Use scientific notation to represent real numbers. Use properties of radicals. Simplify

### Chapter 11 Number Theory

Chapter 11 Number Theory Number theory is one of the oldest branches of mathematics. For many years people who studied number theory delighted in its pure nature because there were few practical applications

### Solving Rational Equations

Lesson M Lesson : Student Outcomes Students solve rational equations, monitoring for the creation of extraneous solutions. Lesson Notes In the preceding lessons, students learned to add, subtract, multiply,

### Answer Key for California State Standards: Algebra I

Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.

### Week 13 Trigonometric Form of Complex Numbers

Week Trigonometric Form of Complex Numbers Overview In this week of the course, which is the last week if you are not going to take calculus, we will look at how Trigonometry can sometimes help in working

### SQUARE-SQUARE ROOT AND CUBE-CUBE ROOT

UNIT 3 SQUAREQUARE AND CUBEUBE (A) Main Concepts and Results A natural number is called a perfect square if it is the square of some natural number. i.e., if m = n 2, then m is a perfect square where m

### ALGEBRA. sequence, term, nth term, consecutive, rule, relationship, generate, predict, continue increase, decrease finite, infinite

ALGEBRA Pupils should be taught to: Generate and describe sequences As outcomes, Year 7 pupils should, for example: Use, read and write, spelling correctly: sequence, term, nth term, consecutive, rule,

### Introduction. Appendix D Mathematical Induction D1

Appendix D Mathematical Induction D D Mathematical Induction Use mathematical induction to prove a formula. Find a sum of powers of integers. Find a formula for a finite sum. Use finite differences to

### 5.1 Radical Notation and Rational Exponents

Section 5.1 Radical Notation and Rational Exponents 1 5.1 Radical Notation and Rational Exponents We now review how exponents can be used to describe not only powers (such as 5 2 and 2 3 ), but also roots

### Review of Fundamental Mathematics

Review of Fundamental Mathematics As explained in the Preface and in Chapter 1 of your textbook, managerial economics applies microeconomic theory to business decision making. The decision-making tools

### Section 3-3 Approximating Real Zeros of Polynomials

- Approimating Real Zeros of Polynomials 9 Section - Approimating Real Zeros of Polynomials Locating Real Zeros The Bisection Method Approimating Multiple Zeros Application The methods for finding zeros

### The thing that started it 8.6 THE BINOMIAL THEOREM

476 Chapter 8 Discrete Mathematics: Functions on the Set of Natural Numbers (b) Based on your results for (a), guess the minimum number of moves required if you start with an arbitrary number of n disks.

### LIES MY CALCULATOR AND COMPUTER TOLD ME

LIES MY CALCULATOR AND COMPUTER TOLD ME See Section Appendix.4 G for a discussion of graphing calculators and computers with graphing software. A wide variety of pocket-size calculating devices are currently

### Section 6-3 Arithmetic and Geometric Sequences

466 6 SEQUENCES, SERIES, AND PROBABILITY Section 6- Arithmetic and Geometric Sequences Arithmetic and Geometric Sequences nth-term Formulas Sum Formulas for Finite Arithmetic Series Sum Formulas for Finite

### Scope and Sequence KA KB 1A 1B 2A 2B 3A 3B 4A 4B 5A 5B 6A 6B

Scope and Sequence Earlybird Kindergarten, Standards Edition Primary Mathematics, Standards Edition Copyright 2008 [SingaporeMath.com Inc.] The check mark indicates where the topic is first introduced

### 3 Some Integer Functions

3 Some Integer Functions A Pair of Fundamental Integer Functions The integer function that is the heart of this section is the modulo function. However, before getting to it, let us look at some very simple

### Grade 7/8 Math Circles Sequences and Series

Faculty of Mathematics Waterloo, Ontario N2L 3G1 Centre for Education in Mathematics and Computing Grade 7/8 Math Circles Sequences and Series November 30, 2012 What are sequences? A sequence is an ordered

### 1 if 1 x 0 1 if 0 x 1

Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or

### POLYNOMIAL FUNCTIONS

POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a

### Taylor and Maclaurin Series

Taylor and Maclaurin Series In the preceding section we were able to find power series representations for a certain restricted class of functions. Here we investigate more general problems: Which functions

### Negative Integral Exponents. If x is nonzero, the reciprocal of x is written as 1 x. For example, the reciprocal of 23 is written as 2

4 (4-) Chapter 4 Polynomials and Eponents P( r) 0 ( r) dollars. Which law of eponents can be used to simplify the last epression? Simplify it. P( r) 7. CD rollover. Ronnie invested P dollars in a -year

### Unit 7 The Number System: Multiplying and Dividing Integers

Unit 7 The Number System: Multiplying and Dividing Integers Introduction In this unit, students will multiply and divide integers, and multiply positive and negative fractions by integers. Students will

### Continued Fractions. Darren C. Collins

Continued Fractions Darren C Collins Abstract In this paper, we discuss continued fractions First, we discuss the definition and notation Second, we discuss the development of the subject throughout history

### 6. Vectors. 1 2009-2016 Scott Surgent (surgent@asu.edu)

6. Vectors For purposes of applications in calculus and physics, a vector has both a direction and a magnitude (length), and is usually represented as an arrow. The start of the arrow is the vector s foot,

### Factoring and Applications

Factoring and Applications What is a factor? The Greatest Common Factor (GCF) To factor a number means to write it as a product (multiplication). Therefore, in the problem 48 3, 4 and 8 are called the

### SECTION 10-2 Mathematical Induction

73 0 Sequences and Series 6. Approximate e 0. using the first five terms of the series. Compare this approximation with your calculator evaluation of e 0.. 6. Approximate e 0.5 using the first five terms

### Math 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.

Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used

### MATH 60 NOTEBOOK CERTIFICATIONS

MATH 60 NOTEBOOK CERTIFICATIONS Chapter #1: Integers and Real Numbers 1.1a 1.1b 1.2 1.3 1.4 1.8 Chapter #2: Algebraic Expressions, Linear Equations, and Applications 2.1a 2.1b 2.1c 2.2 2.3a 2.3b 2.4 2.5

### Activity 1: Using base ten blocks to model operations on decimals

Rational Numbers 9: Decimal Form of Rational Numbers Objectives To use base ten blocks to model operations on decimal numbers To review the algorithms for addition, subtraction, multiplication and division

### Algebra Word Problems

WORKPLACE LINK: Nancy works at a clothing store. A customer wants to know the original price of a pair of slacks that are now on sale for 40% off. The sale price is \$6.50. Nancy knows that 40% of the original

### 0.8 Rational Expressions and Equations

96 Prerequisites 0.8 Rational Expressions and Equations We now turn our attention to rational expressions - that is, algebraic fractions - and equations which contain them. The reader is encouraged to

### Rational Exponents. Squaring both sides of the equation yields. and to be consistent, we must have

8.6 Rational Exponents 8.6 OBJECTIVES 1. Define rational exponents 2. Simplify expressions containing rational exponents 3. Use a calculator to estimate the value of an expression containing rational exponents

### Trigonometric Functions and Triangles

Trigonometric Functions and Triangles Dr. Philippe B. Laval Kennesaw STate University August 27, 2010 Abstract This handout defines the trigonometric function of angles and discusses the relationship between

### Just the Factors, Ma am

1 Introduction Just the Factors, Ma am The purpose of this note is to find and study a method for determining and counting all the positive integer divisors of a positive integer Let N be a given positive

### Mathematics Pre-Test Sample Questions A. { 11, 7} B. { 7,0,7} C. { 7, 7} D. { 11, 11}

Mathematics Pre-Test Sample Questions 1. Which of the following sets is closed under division? I. {½, 1,, 4} II. {-1, 1} III. {-1, 0, 1} A. I only B. II only C. III only D. I and II. Which of the following

### Vocabulary Words and Definitions for Algebra

Name: Period: Vocabulary Words and s for Algebra Absolute Value Additive Inverse Algebraic Expression Ascending Order Associative Property Axis of Symmetry Base Binomial Coefficient Combine Like Terms

### Math Workshop October 2010 Fractions and Repeating Decimals

Math Workshop October 2010 Fractions and Repeating Decimals This evening we will investigate the patterns that arise when converting fractions to decimals. As an example of what we will be looking at,

### MATH 10034 Fundamental Mathematics IV

MATH 0034 Fundamental Mathematics IV http://www.math.kent.edu/ebooks/0034/funmath4.pdf Department of Mathematical Sciences Kent State University January 2, 2009 ii Contents To the Instructor v Polynomials.

### LINEAR EQUATIONS IN TWO VARIABLES

66 MATHEMATICS CHAPTER 4 LINEAR EQUATIONS IN TWO VARIABLES The principal use of the Analytic Art is to bring Mathematical Problems to Equations and to exhibit those Equations in the most simple terms that

### Session 7 Fractions and Decimals

Key Terms in This Session Session 7 Fractions and Decimals Previously Introduced prime number rational numbers New in This Session period repeating decimal terminating decimal Introduction In this session,

### Chapter 9. Systems of Linear Equations

Chapter 9. Systems of Linear Equations 9.1. Solve Systems of Linear Equations by Graphing KYOTE Standards: CR 21; CA 13 In this section we discuss how to solve systems of two linear equations in two variables

### Application. Outline. 3-1 Polynomial Functions 3-2 Finding Rational Zeros of. Polynomial. 3-3 Approximating Real Zeros of.

Polynomial and Rational Functions Outline 3-1 Polynomial Functions 3-2 Finding Rational Zeros of Polynomials 3-3 Approximating Real Zeros of Polynomials 3-4 Rational Functions Chapter 3 Group Activity:

### Figure 1. A typical Laboratory Thermometer graduated in C.

SIGNIFICANT FIGURES, EXPONENTS, AND SCIENTIFIC NOTATION 2004, 1990 by David A. Katz. All rights reserved. Permission for classroom use as long as the original copyright is included. 1. SIGNIFICANT FIGURES

### 4. How many integers between 2004 and 4002 are perfect squares?

5 is 0% of what number? What is the value of + 3 4 + 99 00? (alternating signs) 3 A frog is at the bottom of a well 0 feet deep It climbs up 3 feet every day, but slides back feet each night If it started

### Representation of functions as power series

Representation of functions as power series Dr. Philippe B. Laval Kennesaw State University November 9, 008 Abstract This document is a summary of the theory and techniques used to represent functions

### Cubes and Cube Roots

CUBES AND CUBE ROOTS 109 Cubes and Cube Roots CHAPTER 7 7.1 Introduction This is a story about one of India s great mathematical geniuses, S. Ramanujan. Once another famous mathematician Prof. G.H. Hardy

### If A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C?

Problem 3 If A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C? Suggested Questions to ask students about Problem 3 The key to this question

### ALGEBRA 2/TRIGONOMETRY

ALGEBRA /TRIGONOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION ALGEBRA /TRIGONOMETRY Thursday, January 9, 015 9:15 a.m to 1:15 p.m., only Student Name: School Name: The possession

### Grade 5 Math Content 1

Grade 5 Math Content 1 Number and Operations: Whole Numbers Multiplication and Division In Grade 5, students consolidate their understanding of the computational strategies they use for multiplication.

### SAT Subject Math Level 2 Facts & Formulas

Numbers, Sequences, Factors Integers:..., -3, -2, -1, 0, 1, 2, 3,... Reals: integers plus fractions, decimals, and irrationals ( 2, 3, π, etc.) Order Of Operations: Arithmetic Sequences: PEMDAS (Parentheses

### Zeros of a Polynomial Function

Zeros of a Polynomial Function An important consequence of the Factor Theorem is that finding the zeros of a polynomial is really the same thing as factoring it into linear factors. In this section we

### Teaching & Learning Plans. Arithmetic Sequences. Leaving Certificate Syllabus

Teaching & Learning Plans Arithmetic Sequences Leaving Certificate Syllabus The Teaching & Learning Plans are structured as follows: Aims outline what the lesson, or series of lessons, hopes to achieve.

### Stanford Math Circle: Sunday, May 9, 2010 Square-Triangular Numbers, Pell s Equation, and Continued Fractions

Stanford Math Circle: Sunday, May 9, 00 Square-Triangular Numbers, Pell s Equation, and Continued Fractions Recall that triangular numbers are numbers of the form T m = numbers that can be arranged in

### Undergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics

Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics An Introductory Single Variable Real Analysis: A Learning Approach through Problem Solving Marcel B. Finan c All Rights

### 26 Integers: Multiplication, Division, and Order

26 Integers: Multiplication, Division, and Order Integer multiplication and division are extensions of whole number multiplication and division. In multiplying and dividing integers, the one new issue

### ALGEBRA 2/TRIGONOMETRY

ALGEBRA /TRIGONOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION ALGEBRA /TRIGONOMETRY Tuesday, June 1, 011 1:15 to 4:15 p.m., only Student Name: School Name: Print your name

### Sect 6.7 - Solving Equations Using the Zero Product Rule

Sect 6.7 - Solving Equations Using the Zero Product Rule 116 Concept #1: Definition of a Quadratic Equation A quadratic equation is an equation that can be written in the form ax 2 + bx + c = 0 (referred

### Overview. Essential Questions. Precalculus, Quarter 4, Unit 4.5 Build Arithmetic and Geometric Sequences and Series

Sequences and Series Overview Number of instruction days: 4 6 (1 day = 53 minutes) Content to Be Learned Write arithmetic and geometric sequences both recursively and with an explicit formula, use them

### Free Pre-Algebra Lesson 55! page 1

Free Pre-Algebra Lesson 55! page 1 Lesson 55 Perimeter Problems with Related Variables Take your skill at word problems to a new level in this section. All the problems are the same type, so that you can

### 3.1. RATIONAL EXPRESSIONS

3.1. RATIONAL EXPRESSIONS RATIONAL NUMBERS In previous courses you have learned how to operate (do addition, subtraction, multiplication, and division) on rational numbers (fractions). Rational numbers

### LINEAR INEQUALITIES. Mathematics is the art of saying many things in many different ways. MAXWELL

Chapter 6 LINEAR INEQUALITIES 6.1 Introduction Mathematics is the art of saying many things in many different ways. MAXWELL In earlier classes, we have studied equations in one variable and two variables

### Core Maths C1. Revision Notes

Core Maths C Revision Notes November 0 Core Maths C Algebra... Indices... Rules of indices... Surds... 4 Simplifying surds... 4 Rationalising the denominator... 4 Quadratic functions... 4 Completing the

### Session 29 Scientific Notation and Laws of Exponents. If you have ever taken a Chemistry class, you may have encountered the following numbers:

Session 9 Scientific Notation and Laws of Exponents If you have ever taken a Chemistry class, you may have encountered the following numbers: There are approximately 60,4,79,00,000,000,000,000 molecules

Academic Content Standards Grade Eight and Grade Nine Ohio Algebra 1 2008 Grade Eight STANDARDS Number, Number Sense and Operations Standard Number and Number Systems 1. Use scientific notation to express

### Discrete Mathematics. Chapter 11 Sequences and Series. Chapter 12 Probability and Statistics

Discrete Mathematics Discrete mathematics is the branch of mathematics that involves finite or discontinuous quantities. In this unit, you will learn about sequences, series, probability, and statistics.

### 8 Primes and Modular Arithmetic

8 Primes and Modular Arithmetic 8.1 Primes and Factors Over two millennia ago already, people all over the world were considering the properties of numbers. One of the simplest concepts is prime numbers.

476 CHAPTER 7 Graphs, Equations, and Inequalities 7. Quadratic Equations Now Work the Are You Prepared? problems on page 48. OBJECTIVES 1 Solve Quadratic Equations by Factoring (p. 476) Solve Quadratic

### Consumer Math 15 INDEPENDENT LEAR NING S INC E 1975. Consumer Math

Consumer Math 15 INDEPENDENT LEAR NING S INC E 1975 Consumer Math Consumer Math ENROLLED STUDENTS ONLY This course is designed for the student who is challenged by abstract forms of higher This math. course

### a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.

Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given

### Continued Fractions and the Euclidean Algorithm

Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

### Algebra 2 Chapter 1 Vocabulary. identity - A statement that equates two equivalent expressions.

Chapter 1 Vocabulary identity - A statement that equates two equivalent expressions. verbal model- A word equation that represents a real-life problem. algebraic expression - An expression with variables.

### Second Order Linear Nonhomogeneous Differential Equations; Method of Undetermined Coefficients. y + p(t) y + q(t) y = g(t), g(t) 0.

Second Order Linear Nonhomogeneous Differential Equations; Method of Undetermined Coefficients We will now turn our attention to nonhomogeneous second order linear equations, equations with the standard

### CHAPTER 5. Number Theory. 1. Integers and Division. Discussion

CHAPTER 5 Number Theory 1. Integers and Division 1.1. Divisibility. Definition 1.1.1. Given two integers a and b we say a divides b if there is an integer c such that b = ac. If a divides b, we write a

### Answer: The relationship cannot be determined.

Question 1 Test 2, Second QR Section (version 3) In City X, the range of the daily low temperatures during... QA: The range of the daily low temperatures in City X... QB: 30 Fahrenheit Arithmetic: Ranges

### Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2

CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2 Proofs Intuitively, the concept of proof should already be familiar We all like to assert things, and few of us

### 1.7 Graphs of Functions

64 Relations and Functions 1.7 Graphs of Functions In Section 1.4 we defined a function as a special type of relation; one in which each x-coordinate was matched with only one y-coordinate. We spent most

### To discuss this topic fully, let us define some terms used in this and the following sets of supplemental notes.

INFINITE SERIES SERIES AND PARTIAL SUMS What if we wanted to sum up the terms of this sequence, how many terms would I have to use? 1, 2, 3,... 10,...? Well, we could start creating sums of a finite number

### Core Maths C2. Revision Notes

Core Maths C Revision Notes November 0 Core Maths C Algebra... Polnomials: +,,,.... Factorising... Long division... Remainder theorem... Factor theorem... 4 Choosing a suitable factor... 5 Cubic equations...