Lecture Notes The Fibonacci Sequence page 1
|
|
- Melanie Harrell
- 7 years ago
- Views:
Transcription
1 Lecture Notes The Fibonacci Sequence age De nition: The Fibonacci sequence starts with and and for all other terms in the sequence, we must add the last two terms. F F and for all n, So the rst few terms of the Fibonacci sequence are ; ; ; 3; ; 8; 3; ; 3; ; ; ; : : : The de nition shown above is a recursive one. If we are needed to comute the 00th term of the sequence, we would be forced to comute rst the rst 99 terms in the sequence. So we are naturally interested in nding a formula that enables us to comute the 00th element directly. Such a formula is callled elicit. Like so many things about this sequence, the elicit formula for its nth term is fascinating and surrising. We will derive this formula later. The Fibonacci sequence is named after Leonardo Fibonacci and has very strange and beautiful roerties. of these roerties are connected to the golden mean, ' + : Consider now another sequence, fq n g that is formed by taking the quotients of consecutive term in the Fibonacci sequence. That is, ; ; 3 ; 3 ; 8 ; 3 8 ;... q n + for all natural number n. The decimal resentations of the terms in this sequence show an interesting attern. 3 : 3 : :6 3 8 :6 3 : : A lot The quotients oscillate back and forth and seem to be closer and closer to each other. Amazingly, there is only one number that is inside all of the "swirls" shown on the icture above. These ratios aroach a single number. We call this number the limit of this sequence and we comute its eact value in the samle roblems. De nition: A Fibonacci-tye of a sequence starts with any two real numbers and the rest of the sequence is generated the same way the Fibonacci sequence is. f ; f R and for all n, f n + f n+ f n+ Suose we start with f 3 and f. The rst few terms of this Fibonacci-tye sequence are 3; ; 7; ; ; 6; ; 67; 08; 7; : : : We can de ne oerations on Fibonacci-tye sequences. Consider fa n g and fb n g de ned as follows: fa n g : 3; ; 7; ; 8; 9; 7; 76; 3; 99; : : : fb n g : ; 7; 9; 6; ; ; 66; 07; 73; 80; : : : c Hidegkuti, 03 Last revised: August, 03
2 Lecture Notes The Fibonacci Sequence age We can multily a sequence by a number by multilying each term by that number: fa n g : 6; 8; ; ; 30; ; 8; 3; 6; 30; : : : and the resulting sequence is still Fibonacci-tye. We can also add two sequences by adding them term by term: c n a n + b n c n fa n + b n g : ; ; 6; 7; 3; 70; 3; 83; : : : and the sum is again Fibonacci-tye. determined by its rst two terms. Also, it is very easy to see that every Fibonacci-tye sequence is uniquely These roerties are used when we derive the elicit formula for the nth term of the Fibonacci sequence. c Hidegkuti, 03 Last revised: August, 03
3 Lecture Notes The Fibonacci Sequence age 3 Samle Problems. Solve the equation +.. De ne ' + and. Prove each of the following. a) ' b) ' 3. Find the limit of of the sequence formed from consecutive terms in the Fibonacci sequence. In short, + comute lim. n!. De nition: Two ositive integers are relatively rime if their greatest common divisor is. Prove that any two consecutive terms of the Fibonacci sequence are relatively rime.. Consider a Fibonacci-tye of a sequence with rst term and second term. Is there a value of for which all terms of the sequence fall between 00 and 00? 6. De nition: A geometric sequence is de ned as a; ar; ar ; ar 3 ; ar ; ::::::. The number r is called the common ratio of the sequence because if r 6 0, then r a n+ for all n N. It is clear that a geometric sequence a n is determined by its rst element and common ratio. One great advantage of a geometric sequence over a Fibonacci-tye of a sequence is that there is a very easy elicit formula for the nth term of the sequence: a n ar n : Is there a Fibonacci-tye sequence that is also a geometric series? 7. Consider the geometric sequence de ned by rst element and common ratio r +. Comute the eact value of the 9th term in the sequence. 8. Use results form the revious roblems to nd the elicit formula for the nth term of the Fibonacci sequence. c Hidegkuti, 03 Last revised: August, 03
4 Lecture Notes The Fibonacci Sequence age Solutions - Samle Problems. Solve the equation + Solution. Solution. Using the quadratic formula Comleting the square 0 ; ( ) {z } 0! 0!! De ne ' + a) ' Solution: b) ' Solution: ' and. Prove each of the following. + + We rationalize the radical eression ' ' using its conjugate. + + c Hidegkuti, 03 Last revised: August, 03
5 Lecture Notes The Fibonacci Sequence age 3. Find the limit of of the sequence formed from consecutive terms in the Fibonacci sequence. In short, + comute lim. n! Solution : Let us assume rst that the consecutive terms ; ; 3 ; 3 ; 8 ; 3 8 ; 3 ; 3 ; 3 ; ; ; : : : do aroach a single number. Imagine we are much further into the sequence, after millions and millions of terms. Then these numbers are very close to and thus also very close to each other. Sort of like we are in the Fibonacci sequence: Using more general notation, we arrive to the same conclusion. For very large values of n, + + we are in the Fibonacci sequence: We solve the equation + + multily by + 0 ; We rule out because it is negative, and the sequence clearly aroaches a number above 0:6. other solution, +, the golden mean is the limit. The c Hidegkuti, 03 Last revised: August, 03
6 Lecture Notes The Fibonacci Sequence age 6 Solution : This is the same comutation but this time it is resented with calculus notation. + Let us denote lim by. n! + lim lim n! n! + lim lim n! n! + lim lim n! n! + lim lim n! n! F n+ Fn + F n+ + + Fn lim lim + n! n! ; We rule out because it is negative, and the sequence clearly aroaches a number above 0:6. + other solution, the golden mean is the limit. lim +. n!. De nition: Two ositive integers are relatively rime if their greatest common divisor is. Prove that any two consecutive terms of the Fibonacci sequence are relatively rime. Solution: this is an interesting alication of roofs by contradiction. Suose for a contradiction that there eist two consecutive terms F k and F k+ (for some natural number k) that are not relatively rime. Then there eists a ositive integer d > such that d is a divisor of both F k and F k+. Then there eist and ositive integers such that F k d and F k+ d. We claim that then d is also a divisor of F k. F k + F k F k+ F k F k+ F k d d d ( ) Thus d also divides F k. Net we similarly rove that d is then also a divisor of F k and F k 3 ; and so on, all the way till F. Thus d is a divisor of F. This is imossible because d > and F. This is a contradiction comleting our roof.. Consider a Fibonacci-tye of a sequence with rst term and second term. Is there a value of for which all terms of the sequence fall between 00 and 00? Solution: The Fibonacci sequence vey quickly becomes very large. The question is: how can we ensure that the terms of the sequence do not become large? Consider a Fibonacci-tye sequence with rst term and second term. ; ; + ; + ; 3 + ; + 3; 8 + ; 3 + 8; + 3; : : : : eventually the terms are +. If is ositive, even if tiny, the other art alone, will ensure that the nth term is very large. Thus, if we want the terms to stay small, we need to be negative. This idea generalizes. We want every second term ositive and every other term negative, because two consecutive terms with the same sign guarantee that the terms after that get very large. Suose that a and b are two consecutive terms with the same sign. Then from then on, we have that a; b; a + b; a + b; a + 3b; 3a + b; a + 8b; : : : ; a + + b ; : : : : So we want alternating sings in the sequence That is: ; ; + ; + ; 3 + ; + 3; 8 + ; 3 + 8; + 3; : : : : The c Hidegkuti, 03 Last revised: August, 03
7 Lecture Notes The Fibonacci Sequence age 7 ositive, negative, + ositive, + negative, 3 + ositive, + 3 negative, 8 + ositive, negative, + 3 ositive, and so on. We solve all these inequalities: < 0 + > 0 ) > + < 0 ) < 3 + > 0 ) > < 0 ) < > 0 ) > < 0 ) < > 0 ) > 3 It looks like must be between the values de ned by consecutive terms of the Fibonacci sequence. These rations dislay a strange behavior, they siral around over a smaller and smaller interval (see roblem 3). The only di erence here is that we are looking at instead of +. The ratios in this roblem aroach + the negative recirocal of the golden mean. We rationalize and obtain. This is the only + number that will work for. This sequence will have terms with alternating signs and thus each term will have a smaller absolute value than the revious term. It is an amazing thought that for any other values of, the sequence will reach huge numbers and outgrow any bound. We can resent a bit more formal comutation: denote the sequence by a n : fa n g : ; ; + ; + ; 3 + ; + 3; 8 + ; 3 + 8; + 3; : : : : Notice that for all n 3 where f g is the Fibonacci sequence. a n + f g : ; ; ; 3; ; 8; 3; ; 3; ; ; We want a, a 3, a,... ositive and a ; a ; a 6 ;... negative. For all n, we need a n+ > 0 and a n < 0. a n + and a n+ + + > 0 and + < 0 > < < and < for all n for all n Since these quotiens oscillate around and enclose only a single number, must be that number. Thus lim n! F m+ lim m! F m + c Hidegkuti, 03 Last revised: August, 03
8 Lecture Notes The Fibonacci Sequence age 8 6. De nition: A geometric sequence is de ned as a; ar; ar ; ar 3 ; ar ; ::::::. The number r is called the common ratio of the sequence because if r 6 0, then r a n+ for all n N. It is clear that a geometric sequence a n is determined by its rst element and common ratio. One great advantage of a geometric sequence over a Fibonacci-tye of a sequence is that there is a very easy elicit formula for the nth term of the sequence: a n ar n : Is there a Fibonacci-tye sequence that is also a geometric series? Solution: Let fa n g be a Fibonacci-tye geometric sequence with rst element a and comon ratio r. Then the rst three elements (since geometric) are a; ar; ar Let us assume that a 6 0. (The constant zero sequence is both Fibonacci-tye and geometric, but not very interesting.) The sequence is also Fibonacci-tye and so a + ar ar 0 ar ar a factor out a 0 a r r divide by a 0 r r r ; ( ) At this oint, we are not surrised that we again bumed into the golden mean. will be very useful later on. Suose that a. Then one sequence is Both solutions work, which ; + ; 3 + ; + ; ; : : : is an increasing sequence that grows unbounded. The other sequence is ; ; ; ; ; : : : is a sequence with alternating sings and thus small terms, and the two sequences aear to be conjugates of each other, term by term. The comutation above shows that only r will result in a non-zero Fibonacci-tye sequence. On the other hand, all other such sequences are just constant muliles of these two. All Fibonacci-tye sequences with rst term a are of the form +! 3 +! a; a ; a ; a ! ; a ; : : : and!! 3 7 3! a; a ; a ; a ; a ; : : : What makes these sequences secial is that their nth term can be so easily determined because they are geometric sequences as well. 7. Consider the geometric sequence de ned by rst element and common ratio r +. Comute the eact value of the 9th term in the sequence. Solution: a 9 ar 8 +! 8 +! 8 c Hidegkuti, 03 Last revised: August, 03
9 Lecture Notes The Fibonacci Sequence age 9 We start with +! +! We square this number: +! +! 3 3 +! We square again: +! 8 +! 3! and so a This might seem laborous but if n is large, it is still much better than having to comute all revious terms. 8. Use results form the revious roblems to nd the elicit formula for the nth term of the Fibonacci sequence. Solution: We will eress the Fibonacci sequence as the sum of two Fibonacci-tye geometric sequences. First, we need to verify that the constant mutlies and sums of Fibonacci-tye sequences are still Fibonaccitye. Second, we will use the fact that the rst two elements uniquely determine any Fibonaccy-tye of sequence. De ne fa n g and fb n g geometric sequences as follows. a and r a + Thus a n +! n and b n! n and b n and r b. These sequences are also Fibonacci-tye (see roblem 6). Thus, any constant multiles and sums formed from these sequences will still be Fibonacci-tye. Could we use fa n g and fb n g to "concoct" the Fibonacci sequence? Let and y be real numbers such that for all n c n a n a n and c and c If we could nd such and y, we would be done because a Fibonacci-tye sequence that begins with and is THE bonacci-sequence. c a a ) +!! c a a ) +!! c Hidegkuti, 03 Last revised: August, 03
10 Lecture Notes The Fibonacci Sequence age 0 8 >< +!! The system >: +!! it. We simlify both equations. Since is a linear system in and y so we should be able to solve +! the system is 8>< >: +! 3 +! 3!! Let us multily by + We solve for y in the rst equation: y + and substitute into the second equation and solve for : multily by add subtract 6 factor out divide by We substitute this into the eression eressing y. y + and c Hidegkuti, 03 Last revised: August, 03
11 Lecture Notes The Fibonacci Sequence age y +! + 0 So and y Consequently, the nth term in the Fibonacci sequence is 0! ! n! n! This formula seems very unlikely to roduce integers. Let us see the rst few elements generated by the formula. F +!!! !! A 3 + 3! Before comuting F 3, let us comute +! 3. +! 3 +! +! +! 3 +! We similarly obtain the eact value of 0 F +! 3! 3 and then we are ready to comute F 3.! 3 A + For more documents like this, visit our age at htt:// and click on Lecture Notes. Our goal is to emower students to learn and enjoy mathematics free of charge. If you have any questions or comments, to mhidegkuti@ccc.edu. c Hidegkuti, 03 Last revised: August, 03
Pythagorean Triples and Rational Points on the Unit Circle
Pythagorean Triles and Rational Points on the Unit Circle Solutions Below are samle solutions to the roblems osed. You may find that your solutions are different in form and you may have found atterns
More informationSample Problems. Practice Problems
Lecture Notes Quadratic Word Problems page 1 Sample Problems 1. The sum of two numbers is 31, their di erence is 41. Find these numbers.. The product of two numbers is 640. Their di erence is 1. Find these
More information6.042/18.062J Mathematics for Computer Science December 12, 2006 Tom Leighton and Ronitt Rubinfeld. Random Walks
6.042/8.062J Mathematics for Comuter Science December 2, 2006 Tom Leighton and Ronitt Rubinfeld Lecture Notes Random Walks Gambler s Ruin Today we re going to talk about one-dimensional random walks. In
More informationMore Properties of Limits: Order of Operations
math 30 day 5: calculating its 6 More Proerties of Limits: Order of Oerations THEOREM 45 (Order of Oerations, Continued) Assume that!a f () L and that m and n are ositive integers Then 5 (Power)!a [ f
More information1 Gambler s Ruin Problem
Coyright c 2009 by Karl Sigman 1 Gambler s Ruin Problem Let N 2 be an integer and let 1 i N 1. Consider a gambler who starts with an initial fortune of $i and then on each successive gamble either wins
More informationSample Problems. Practice Problems
Lecture Notes Circles - Part page Sample Problems. Find an equation for the circle centered at (; ) with radius r = units.. Graph the equation + + = ( ).. Consider the circle ( ) + ( + ) =. Find all points
More informationComplex Conjugation and Polynomial Factorization
Comlex Conjugation and Polynomial Factorization Dave L. Renfro Summer 2004 Central Michigan University I. The Remainder Theorem Let P (x) be a olynomial with comlex coe cients 1 and r be a comlex number.
More informationThis is a square root. The number under the radical is 9. (An asterisk * means multiply.)
Page of Review of Radical Expressions and Equations Skills involving radicals can be divided into the following groups: Evaluate square roots or higher order roots. Simplify radical expressions. Rationalize
More informationFibonacci Numbers and Greatest Common Divisors. The Finonacci numbers are the numbers in the sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144,...
Fibonacci Numbers and Greatest Common Divisors The Finonacci numbers are the numbers in the sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144,.... After starting with two 1s, we get each Fibonacci number
More informationStat 134 Fall 2011: Gambler s ruin
Stat 134 Fall 2011: Gambler s ruin Michael Lugo Setember 12, 2011 In class today I talked about the roblem of gambler s ruin but there wasn t enough time to do it roerly. I fear I may have confused some
More informationThis unit will lay the groundwork for later units where the students will extend this knowledge to quadratic and exponential functions.
Algebra I Overview View unit yearlong overview here Many of the concepts presented in Algebra I are progressions of concepts that were introduced in grades 6 through 8. The content presented in this course
More information36 CHAPTER 1. LIMITS AND CONTINUITY. Figure 1.17: At which points is f not continuous?
36 CHAPTER 1. LIMITS AND CONTINUITY 1.3 Continuity Before Calculus became clearly de ned, continuity meant that one could draw the graph of a function without having to lift the pen and pencil. While this
More informationContinued 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
More informationPartial Fractions. Combining fractions over a common denominator is a familiar operation from algebra:
Partial Fractions Combining fractions over a common denominator is a familiar operation from algebra: From the standpoint of integration, the left side of Equation 1 would be much easier to work with than
More informationReal Roots of Univariate Polynomials with Real Coefficients
Real Roots of Univariate Polynomials with Real Coefficients mostly written by Christina Hewitt March 22, 2012 1 Introduction Polynomial equations are used throughout mathematics. When solving polynomials
More informationRational 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
More informationCHAPTER 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
More informationsin(x) < x sin(x) x < tan(x) sin(x) x cos(x) 1 < sin(x) sin(x) 1 < 1 cos(x) 1 cos(x) = 1 cos2 (x) 1 + cos(x) = sin2 (x) 1 < x 2
. Problem Show that using an ɛ δ proof. sin() lim = 0 Solution: One can see that the following inequalities are true for values close to zero, both positive and negative. This in turn implies that On the
More informationTRANSCENDENTAL NUMBERS
TRANSCENDENTAL NUMBERS JEREMY BOOHER. Introduction The Greeks tried unsuccessfully to square the circle with a comass and straightedge. In the 9th century, Lindemann showed that this is imossible by demonstrating
More information8 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.
More informationSection 1.1 Real Numbers
. Natural numbers (N):. Integer numbers (Z): Section. Real Numbers Types of Real Numbers,, 3, 4,,... 0, ±, ±, ±3, ±4, ±,... REMARK: Any natural number is an integer number, but not any integer number is
More informationDefinition 8.1 Two inequalities are equivalent if they have the same solution set. Add or Subtract the same value on both sides of the inequality.
8 Inequalities Concepts: Equivalent Inequalities Linear and Nonlinear Inequalities Absolute Value Inequalities (Sections 4.6 and 1.1) 8.1 Equivalent Inequalities Definition 8.1 Two inequalities are equivalent
More informationRepresentation 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
More informationPRIME NUMBERS AND THE RIEMANN HYPOTHESIS
PRIME NUMBERS AND THE RIEMANN HYPOTHESIS CARL ERICKSON This minicourse has two main goals. The first is to carefully define the Riemann zeta function and exlain how it is connected with the rime numbers.
More informationCollatz Sequence. Fibbonacci Sequence. n is even; Recurrence Relation: a n+1 = a n + a n 1.
Fibonacci Roulette In this game you will be constructing a recurrence relation, that is, a sequence of numbers where you find the next number by looking at the previous numbers in the sequence. Your job
More information2.1 Increasing, Decreasing, and Piecewise Functions; Applications
2.1 Increasing, Decreasing, and Piecewise Functions; Applications Graph functions, looking for intervals on which the function is increasing, decreasing, or constant, and estimate relative maxima and minima.
More informationMATH 095, College Prep Mathematics: Unit Coverage Pre-algebra topics (arithmetic skills) offered through BSE (Basic Skills Education)
MATH 095, College Prep Mathematics: Unit Coverage Pre-algebra topics (arithmetic skills) offered through BSE (Basic Skills Education) Accurately add, subtract, multiply, and divide whole numbers, integers,
More informationPrecalculus Prerequisites a.k.a. Chapter 0. August 16, 2013
Precalculus Prerequisites a.k.a. Chater 0 by Carl Stitz, Ph.D. Lakeland Community College Jeff Zeager, Ph.D. Lorain County Community College August 6, 0 Table of Contents 0 Prerequisites 0. Basic Set
More informationAssignment 5 - Due Friday March 6
Assignment 5 - Due Friday March 6 (1) Discovering Fibonacci Relationships By experimenting with numerous examples in search of a pattern, determine a simple formula for (F n+1 ) 2 + (F n ) 2 that is, a
More informationMath 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
More informationMath 55: Discrete Mathematics
Math 55: Discrete Mathematics UC Berkeley, Spring 2012 Homework # 9, due Wednesday, April 11 8.1.5 How many ways are there to pay a bill of 17 pesos using a currency with coins of values of 1 peso, 2 pesos,
More informationMath Common Core Sampler Test
High School Algebra Core Curriculum Math Test Math Common Core Sampler Test Our High School Algebra sampler covers the twenty most common questions that we see targeted for this level. For complete tests
More informationZeros of Polynomial Functions
Zeros of Polynomial Functions The Rational Zero Theorem If f (x) = a n x n + a n-1 x n-1 + + a 1 x + a 0 has integer coefficients and p/q (where p/q is reduced) is a rational zero, then p is a factor of
More informationEstimated Pre Calculus Pacing Timeline
Estimated Pre Calculus Pacing Timeline 2010-2011 School Year The timeframes listed on this calendar are estimates based on a fifty-minute class period. You may need to adjust some of them from time to
More informationProperties 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
More informationMethod To Solve Linear, Polynomial, or Absolute Value Inequalities:
Solving Inequalities An inequality is the result of replacing the = sign in an equation with ,, or. For example, 3x 2 < 7 is a linear inequality. We call it linear because if the < were replaced with
More informationLecture 21 and 22: The Prime Number Theorem
Lecture and : The Prime Number Theorem (New lecture, not in Tet) The location of rime numbers is a central question in number theory. Around 88, Legendre offered eerimental evidence that the number π()
More informationIntroduction. 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
More informationn 2 + 4n + 3. The answer in decimal form (for the Blitz): 0, 75. Solution. (n + 1)(n + 3) = n + 3 2 lim m 2 1
. Calculate the sum of the series Answer: 3 4. n 2 + 4n + 3. The answer in decimal form (for the Blitz):, 75. Solution. n 2 + 4n + 3 = (n + )(n + 3) = (n + 3) (n + ) = 2 (n + )(n + 3) ( 2 n + ) = m ( n
More information1.2 Solving a System of Linear Equations
1.. SOLVING A SYSTEM OF LINEAR EQUATIONS 1. Solving a System of Linear Equations 1..1 Simple Systems - Basic De nitions As noticed above, the general form of a linear system of m equations in n variables
More informationPrice Elasticity of Demand MATH 104 and MATH 184 Mark Mac Lean (with assistance from Patrick Chan) 2011W
Price Elasticity of Demand MATH 104 and MATH 184 Mark Mac Lean (with assistance from Patrick Chan) 2011W The rice elasticity of demand (which is often shortened to demand elasticity) is defined to be the
More informationParamedic Program Pre-Admission Mathematics Test Study Guide
Paramedic Program Pre-Admission Mathematics Test Study Guide 05/13 1 Table of Contents Page 1 Page 2 Page 3 Page 4 Page 5 Page 6 Page 7 Page 8 Page 9 Page 10 Page 11 Page 12 Page 13 Page 14 Page 15 Page
More information3 0 + 4 + 3 1 + 1 + 3 9 + 6 + 3 0 + 1 + 3 0 + 1 + 3 2 mod 10 = 4 + 3 + 1 + 27 + 6 + 1 + 1 + 6 mod 10 = 49 mod 10 = 9.
SOLUTIONS TO HOMEWORK 2 - MATH 170, SUMMER SESSION I (2012) (1) (Exercise 11, Page 107) Which of the following is the correct UPC for Progresso minestrone soup? Show why the other numbers are not valid
More informationCopy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any.
Algebra 2 - Chapter Prerequisites Vocabulary Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. P1 p. 1 1. counting(natural) numbers - {1,2,3,4,...}
More informationLies 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
More informationMath 115 Spring 2011 Written Homework 5 Solutions
. Evaluate each series. a) 4 7 0... 55 Math 5 Spring 0 Written Homework 5 Solutions Solution: We note that the associated sequence, 4, 7, 0,..., 55 appears to be an arithmetic sequence. If the sequence
More informationLecture 2. Marginal Functions, Average Functions, Elasticity, the Marginal Principle, and Constrained Optimization
Lecture 2. Marginal Functions, Average Functions, Elasticity, the Marginal Principle, and Constrained Optimization 2.1. Introduction Suppose that an economic relationship can be described by a real-valued
More informationExamples of Functions
Examples of Functions In this document is provided examples of a variety of functions. The purpose is to convince the beginning student that functions are something quite different than polynomial equations.
More informationLectures on the Dirichlet Class Number Formula for Imaginary Quadratic Fields. Tom Weston
Lectures on the Dirichlet Class Number Formula for Imaginary Quadratic Fields Tom Weston Contents Introduction 4 Chater 1. Comlex lattices and infinite sums of Legendre symbols 5 1. Comlex lattices 5
More informationLecture Notes Order of Operations page 1
Lecture Notes Order of Operations page 1 The order of operations rule is an agreement among mathematicians, it simpli es notation. P stands for parentheses, E for exponents, M and D for multiplication
More information1.7 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
More informationUndergraduate 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
More informationCONTINUED FRACTIONS AND PELL S EQUATION. Contents 1. Continued Fractions 1 2. Solution to Pell s Equation 9 References 12
CONTINUED FRACTIONS AND PELL S EQUATION SEUNG HYUN YANG Abstract. In this REU paper, I will use some important characteristics of continued fractions to give the complete set of solutions to Pell s equation.
More informationHow To Understand And Solve Algebraic Equations
College Algebra Course Text Barnett, Raymond A., Michael R. Ziegler, and Karl E. Byleen. College Algebra, 8th edition, McGraw-Hill, 2008, ISBN: 978-0-07-286738-1 Course Description This course provides
More informationCommon sense, and the model that we have used, suggest that an increase in p means a decrease in demand, but this is not the only possibility.
Lecture 6: Income and Substitution E ects c 2009 Je rey A. Miron Outline 1. Introduction 2. The Substitution E ect 3. The Income E ect 4. The Sign of the Substitution E ect 5. The Total Change in Demand
More informationMath 181 Handout 16. Rich Schwartz. March 9, 2010
Math 8 Handout 6 Rich Schwartz March 9, 200 The purpose of this handout is to describe continued fractions and their connection to hyperbolic geometry. The Gauss Map Given any x (0, ) we define γ(x) =
More informationMATH 0110 Developmental Math Skills Review, 1 Credit, 3 hours lab
MATH 0110 Developmental Math Skills Review, 1 Credit, 3 hours lab MATH 0110 is established to accommodate students desiring non-course based remediation in developmental mathematics. This structure will
More informationRandom variables, probability distributions, binomial random variable
Week 4 lecture notes. WEEK 4 page 1 Random variables, probability distributions, binomial random variable Eample 1 : Consider the eperiment of flipping a fair coin three times. The number of tails that
More informationSection 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
More information1 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
More informationChapter 13: Fibonacci Numbers and the Golden Ratio
Chapter 13: Fibonacci Numbers and the Golden Ratio 13.1 Fibonacci Numbers THE FIBONACCI SEQUENCE 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, The sequence of numbers shown above is called the Fibonacci
More informationSequences. A sequence is a list of numbers, or a pattern, which obeys a rule.
Sequences A sequence is a list of numbers, or a pattern, which obeys a rule. Each number in a sequence is called a term. ie the fourth term of the sequence 2, 4, 6, 8, 10, 12... is 8, because it is the
More informationMath 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,
More informationHOMEWORK 5 SOLUTIONS. n!f n (1) lim. ln x n! + xn x. 1 = G n 1 (x). (2) k + 1 n. (n 1)!
Math 7 Fall 205 HOMEWORK 5 SOLUTIONS Problem. 2008 B2 Let F 0 x = ln x. For n 0 and x > 0, let F n+ x = 0 F ntdt. Evaluate n!f n lim n ln n. By directly computing F n x for small n s, we obtain the following
More informationStudent Outcomes. Lesson Notes. Classwork. Discussion (10 minutes)
NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 5 8 Student Outcomes Students know the definition of a number raised to a negative exponent. Students simplify and write equivalent expressions that contain
More informationMath 55: Discrete Mathematics
Math 55: Discrete Mathematics UC Berkeley, Fall 2011 Homework # 5, due Wednesday, February 22 5.1.4 Let P (n) be the statement that 1 3 + 2 3 + + n 3 = (n(n + 1)/2) 2 for the positive integer n. a) What
More informationLinear Equations and Inequalities
Linear Equations and Inequalities Section 1.1 Prof. Wodarz Math 109 - Fall 2008 Contents 1 Linear Equations 2 1.1 Standard Form of a Linear Equation................ 2 1.2 Solving Linear Equations......................
More informationDRAFT. Algebra 1 EOC Item Specifications
DRAFT Algebra 1 EOC Item Specifications The draft Florida Standards Assessment (FSA) Test Item Specifications (Specifications) are based upon the Florida Standards and the Florida Course Descriptions as
More informationApplication. 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:
More informationVocabulary 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
More informationCreating, Solving, and Graphing Systems of Linear Equations and Linear Inequalities
Algebra 1, Quarter 2, Unit 2.1 Creating, Solving, and Graphing Systems of Linear Equations and Linear Inequalities Overview Number of instructional days: 15 (1 day = 45 60 minutes) Content to be learned
More informationMath 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
More informationSouth Carolina College- and Career-Ready (SCCCR) Algebra 1
South Carolina College- and Career-Ready (SCCCR) Algebra 1 South Carolina College- and Career-Ready Mathematical Process Standards The South Carolina College- and Career-Ready (SCCCR) Mathematical Process
More informationCore 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
More informationSolving 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,
More informationSome Polynomial Theorems. John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 rkennedy@ix.netcom.
Some Polynomial Theorems by John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 rkennedy@ix.netcom.com This paper contains a collection of 31 theorems, lemmas,
More informationAPPLICATIONS 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
More information8 Divisibility and prime numbers
8 Divisibility and prime numbers 8.1 Divisibility In this short section we extend the concept of a multiple from the natural numbers to the integers. We also summarize several other terms that express
More informationFlorida Math 0028. Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies - Upper
Florida Math 0028 Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies - Upper Exponents & Polynomials MDECU1: Applies the order of operations to evaluate algebraic
More informationA.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
More informationNumber Theory Naoki Sato <ensato@hotmail.com>
Number Theory Naoki Sato 0 Preface This set of notes on number theory was originally written in 1995 for students at the IMO level. It covers the basic background material that an
More informationAssignment 9; Due Friday, March 17
Assignment 9; Due Friday, March 17 24.4b: A icture of this set is shown below. Note that the set only contains oints on the lines; internal oints are missing. Below are choices for U and V. Notice that
More informationAs we have seen, there is a close connection between Legendre symbols of the form
Gauss Sums As we have seen, there is a close connection between Legendre symbols of the form 3 and cube roots of unity. Secifically, if is a rimitive cube root of unity, then 2 ± i 3 and hence 2 2 3 In
More informationSECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS
(Section 0.6: Polynomial, Rational, and Algebraic Expressions) 0.6.1 SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS LEARNING OBJECTIVES Be able to identify polynomial, rational, and algebraic
More informationElements of probability theory
2 Elements of probability theory Probability theory provides mathematical models for random phenomena, that is, phenomena which under repeated observations yield di erent outcomes that cannot be predicted
More informationCommon Core Standards for Fantasy Sports Worksheets. Page 1
Scoring Systems Concept(s) Integers adding and subtracting integers; multiplying integers Fractions adding and subtracting fractions; multiplying fractions with whole numbers Decimals adding and subtracting
More information9.3 OPERATIONS WITH RADICALS
9. Operations with Radicals (9 1) 87 9. OPERATIONS WITH RADICALS In this section Adding and Subtracting Radicals Multiplying Radicals Conjugates In this section we will use the ideas of Section 9.1 in
More informationJUST THE MATHS UNIT NUMBER 1.8. ALGEBRA 8 (Polynomials) A.J.Hobson
JUST THE MATHS UNIT NUMBER 1.8 ALGEBRA 8 (Polynomials) by A.J.Hobson 1.8.1 The factor theorem 1.8.2 Application to quadratic and cubic expressions 1.8.3 Cubic equations 1.8.4 Long division of polynomials
More informationCONTINUED FRACTIONS AND FACTORING. Niels Lauritzen
CONTINUED FRACTIONS AND FACTORING Niels Lauritzen ii NIELS LAURITZEN DEPARTMENT OF MATHEMATICAL SCIENCES UNIVERSITY OF AARHUS, DENMARK EMAIL: niels@imf.au.dk URL: http://home.imf.au.dk/niels/ Contents
More information5.1 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
More informationMATH 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
More informationThe degree of a polynomial function is equal to the highest exponent found on the independent variables.
DETAILED SOLUTIONS AND CONCEPTS - POLYNOMIAL FUNCTIONS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you! PLEASE NOTE
More information4/1/2017. 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
More informationSection 6-3 Double-Angle and Half-Angle Identities
6-3 Double-Angle and Half-Angle Identities 47 Section 6-3 Double-Angle and Half-Angle Identities Double-Angle Identities Half-Angle Identities This section develops another important set of identities
More informationIB 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 =
More informationLAKE ELSINORE UNIFIED SCHOOL DISTRICT
LAKE ELSINORE UNIFIED SCHOOL DISTRICT Title: PLATO Algebra 1-Semester 2 Grade Level: 10-12 Department: Mathematics Credit: 5 Prerequisite: Letter grade of F and/or N/C in Algebra 1, Semester 2 Course Description:
More informationMicroeconomic Theory: Basic Math Concepts
Microeconomic Theory: Basic Math Concepts Matt Van Essen University of Alabama Van Essen (U of A) Basic Math Concepts 1 / 66 Basic Math Concepts In this lecture we will review some basic mathematical concepts
More information1.2. 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
More informationZeros of Polynomial Functions
Review: Synthetic Division Find (x 2-5x - 5x 3 + x 4 ) (5 + x). Factor Theorem Solve 2x 3-5x 2 + x + 2 =0 given that 2 is a zero of f(x) = 2x 3-5x 2 + x + 2. Zeros of Polynomial Functions Introduction
More informationExponents and Radicals
Exponents and Radicals (a + b) 10 Exponents are a very important part of algebra. An exponent is just a convenient way of writing repeated multiplications of the same number. Radicals involve the use of
More information