The Fibonacci Numbers
|
|
- Priscilla Lee
- 7 years ago
- Views:
Transcription
1 The Fibonacci Numbers The numbers are: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, Each Fibonacci number is the sum of the previous two Fibonacci numbers! Let n any positive integer If F n is what we use to describe the n th Fibonacci number, then F n = F n 1 + F n 2
2 Trying to prove: F 1 + F F n = F n+2 1 We proved a theorem about the sum of the first few Fibonacci numbers: Theorem For any positive integer n, the Fibonacci numbers satisfy: F 1 + F 2 + F F n = F n+2 1
3 The even Fibonacci Numbers What about the first few Fibonacci numbers with even index: F 2, F 4, F 6,, F 2n, Let s call them even Fibonaccis, since their index is even, although the numbers themselves aren t always even!!
4 The even Fibonacci Numbers Some notation: The first even Fibonacci number is F 2 = 1 The second even Fibonacci number is F 4 = 3 The third even Fibonacci number is F 6 = 8 The tenth even Fibonacci number is F 20 =?? The n th even Fibonacci number is F 2n
5 HOMEWORK Tonight, try to come up with a formula for the sum of the first few even Fibonacci numbers
6 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, Let s look at the first few: F 2 + F 4 = = 4 F 2 + F 4 + F 6 = = 12 F 2 + F 4 + F 6 + F 8 = = 33 F 2 + F 4 + F 6 + F 8 + F 10 = = 88 See a pattern?
7 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, It looks like the sum of the first few even Fibonacci numbers is one less than another Fibonacci number But which one? F 2 + F 4 = = 4 = F 5 1 F 2 + F 4 + F 6 = = 12 = F 7 1 F 2 + F 4 + F 6 + F 8 = = 33 = F 9 1 F 2 + F 4 + F 6 + F 8 + F 10 = = 88 = F 11 1 Can we come up with a formula for the sum of the first few even Fibonacci numbers?
8 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, Let s look at the last one we figured out: F 2 + F 4 + F 6 + F 8 + F 10 = = 88 = F 11 1 The sum of the first 5 even Fibonacci numbers (up to F 10 ) is the 11 th Fibonacci number less one Maybe it s true that the sum of the first n even Fibonacci s is one less than the next Fibonacci number That is, Conjecture For any positive integer n, the Fibonacci numbers satisfy: F 2 + F 4 + F F 2n = F 2n+1 1
9 Trying to prove: F 2 + F 4 + F 2n = F 2n+1 1 Let s try to prove this!! We re trying to find out what is equal to F 2 + F 4 + F 2n Well, let s proceed like the last theorem
10 Trying to prove: F 2 + F 4 + F 2n = F 2n+1 1 We know F 3 = F 2 + F 1 So, rewriting this a little: F 2 = F 3 F 1 Also: we know F 5 = F 4 + F 3 So: F 4 = F 5 F 3 In general: Let s put these together: F even = F next odd F previous odd
11 Trying to prove: F 2 + F F 2n = F 2n+1 1 F 2 = F 3 F 1 F 4 = F 5 F 3 F 6 = F 7 F 5 F 8 = F 9 F 7 F 2n 2 = F 2n 1 F 2n 3 F 2n = F 2n+1 F 2n 1 Adding up all the terms on the left sides will give us something equal to the sum of the terms on the right sides
12 Trying to prove: F 2 + F F 2n = F 2n+1 1 F 2 = /// F 3 F 1 F 4 = F 5 /// F 3 F 6 = F 7 F 5 F 8 = F 9 F 7 F 2n 2 = F 2n 1 F 2n 3 + F 2n = F 2n+1 F 2n 1
13 Trying to prove: F 2 + F F 2n = F 2n+1 1 F 2 = /// F 3 F 1 F 4 = /// F 5 /// F 3 F 6 = F 7 /// F 5 F 8 = F 9 F 7 F 2n 2 = F 2n 1 F 2n 3 + F 2n = F 2n+1 F 2n 1
14 Trying to prove: F 2 + F F 2n = F 2n+1 1 F 2 = /// F 3 F 1 F 4 = /// F 5 /// F 3 F 6 = /// F 7 /// F 5 F 8 = F 9 /// F 7 F 2n 2 = F 2n 1 F 2n 3 + F 2n = F 2n+1 F 2n 1
15 Trying to prove: F 2 + F F 2n = F 2n+1 1 F 2 = /// F 3 F 1 F 4 = /// F 5 /// F 3 F 6 = /// F 7 /// F 5 F 8 = /// F 9 /// F 7 F 2n 2 = F 2n 1 //////// F 2n 3 + F 2n = F 2n+1 F 2n 1
16 Trying to prove: F 2 + F F 2n = F 2n+1 1 F 2 = /// F 3 F 1 F 4 = /// F 5 /// F 3 F 6 = /// F 7 /// F 5 F 8 = /// F 9 /// F 7 F 2n 2 = //////// F 2n 1 //////// F 2n 3 + F 2n = F 2n+1 //////// F 2n 1
17 Trying to prove: F 2 + F F 2n = F 2n+1 1 F 2 = /// F 3 F 1 F 4 = /// F 5 /// F 3 F 6 = /// F 7 /// F 5 F 8 = /// F 9 /// F 7 F 2n 2 = //////// F 2n 1 //////// F 2n 3 + F 2n = F 2n+1 //////// F 2n 1 F 2 + F 4 + F F 2n = F 2n+1 F 1
18 Trying to prove: F 2 + F F 2n = F 2n+1 1 F 2 = /// F 3 F 1 F 4 = /// F 5 /// F 3 F 6 = /// F 7 /// F 5 F 8 = /// F 9 /// F 7 F 2n 2 = //////// F 2n 1 //////// F 2n 3 + F 2n = F 2n+1 //////// F 2n 1 F 2 + F 4 + F F 2n = F 2n+1 1
19 Trying to prove: F 2 + F 4 + F 2n = F 2n+1 1 This proves our second theorem!! Theorem For any positive integer n, the even Fibonacci numbers satisfy: F 2 + F 4 + F F 2n = F 2n+1 1
20 The odd Fibonacci Numbers What about the first few Fibonacci numbers with odd index: F 1, F 3, F 5,, F 2n 1, Let s call them odd Fibonaccis, since their index is odd, although the numbers themselves aren t always odd!!
21 The odd Fibonacci Numbers Some notation: The first odd Fibonacci number is F 1 = 1 The second odd Fibonacci number is F 3 = 2 The third odd Fibonacci number is F 5 = 5 The tenth odd Fibonacci number is F 19 =?? The n th odd Fibonacci number is F 2n 1
22 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, Let s look at the first few: F 1 + F 3 = = 3 F 1 + F 3 + F 5 = = 8 F 1 + F 3 + F 5 + F 7 = = 21 F 1 + F 3 + F 5 + F 7 + F 9 = = 55 See a pattern?
23 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, It looks like the sum of the first few odd Fibonacci numbers is another Fibonacci number But which one? F 1 + F 3 = = 3 = F 4 F 1 + F 3 + F 5 = = 8 = F 6 F 1 + F 3 + F 5 + F 7 = = 21 = F 8 F 1 + F 3 + F 5 + F 7 + F 9 = = 55 = F 10 Can we come up with a formula for the sum of the first few even Fibonacci numbers?
24 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, Maybe it s true that the sum of the first n odd Fibonacci s is the next Fibonacci number That is, Conjecture For any positive integer n, the Fibonacci numbers satisfy: F 1 + F 3 + F F 2n 1 = F 2n
25 Trying to prove: F 1 + F F 2n 1 = F 2n Let s try to prove this!! We know F 2 = F 1 So, rewriting this a little: F 1 = F 2 Also: we know F 4 = F 3 + F 2 So: F 3 = F 4 F 2 In general: F odd = F next even F previous even
26 Trying to prove: F 1 + F F 2n 1 = F 2n F 1 = F 2 F 3 = F 4 F 2 F 5 = F 6 F 4 F 7 = F 8 F 6 F 2n 3 = F 2n 2 F 2n 4 + F 2n 1 = F 2n F 2n 2 Adding up all the terms on the left sides will give us something equal to the sum of the terms on the right sides
27 Trying to prove: F 1 + F F 2n 1 = F 2n F 1 = /// F 2 F 3 = F 4 /// F 2 F 5 = F 6 F 4 F 7 = F 8 F 6 F 2n 3 = F 2n 2 F 2n 4 + F 2n 1 = F 2n F 2n 2
28 Trying to prove: F 1 + F F 2n 1 = F 2n F 1 = /// F 2 F 3 = /// F 4 /// F 2 F 5 = F 6 /// F 4 F 7 = F 8 F 6 F 2n 3 = F 2n 2 F 2n 4 + F 2n 1 = F 2n F 2n 2
29 Trying to prove: F 1 + F F 2n 1 = F 2n F 1 = /// F 2 F 3 = /// F 4 /// F 2 F 5 = /// F 6 /// F 4 F 7 = F 8 /// F 6 F 2n 3 = F 2n 2 F 2n 4 + F 2n 1 = F 2n F 2n 2
30 Trying to prove: F 1 + F F 2n 1 = F 2n F 1 = /// F 2 F 3 = /// F 4 /// F 2 F 5 = /// F 6 /// F 4 F 7 = /// F 8 /// F 6 F 2n 3 = F 2n 2 //////// F 2n 4 + F 2n 1 = F 2n F 2n 2
31 Trying to prove: F 1 + F F 2n 1 = F 2n F 1 = /// F 2 F 3 = /// F 4 /// F 2 F 5 = /// F 6 /// F 4 F 7 = /// F 8 /// F 6 F 2n 3 = //////// F 2n 2 //////// F 2n 4 + F 2n 1 = F 2n //////// F 2n 2 F 1 + F F 2n 1 = F 2n
32 Trying to prove: F 1 + F F 2n 1 = F 2n This proves our third theorem!! Theorem For any positive integer n, the Fibonacci numbers satisfy: F 1 + F 3 + F F 2n 1 = F 2n
33 So far: Theorem (Sum of first few Fibonacci numbers) For any positive integer n, the Fibonacci numbers satisfy: F 1 + F 2 + F F n = F n+2 1 Theorem (Sum of first few EVEN Fibonacci numbers) For any positive integer n, the Fibonacci numbers satisfy: F 2 + F 4 + F F 2n = F 2n+1 1 Theorem (Sum of first few ODD Fibonacci numbers) For any positive integer n, the Fibonacci numbers satisfy: F 1 + F 3 + F F 2n 1 = F 2n
34 One more theorem about sums: Theorem (Sum of first few SQUARES of Fibonacci numbers) For any positive integer n, the Fibonacci numbers satisfy: F F F F 2 n = F n F n+1
35 Trying to prove: F F F F 2 n = F n F n+1 We ll take advantage of the following fact: For each Fibonacci number F n F n+1 = F n + F n 1 F n = F n+1 F n 1 F 2 n = F n F n = F n (F n+1 F n 1 ) F 2 n = F n F n+1 F n F n 1
36 Trying to prove: F F F F 2 n = F n F n+1 Let s check this formula for a few different valus of n: F 2 n = F n F n+1 F n F n 1 n = 4: F 2 4 = F 4 F 4+1 F 4 F 4 1 F 2 4 = F 4 F 5 F 4 F = = 15 6 It works for n = 4!
37 Trying to prove: F F F F 2 n = F n F n+1 Let s check this formula for a few different valus of n: F 2 n = F n F n+1 F n F n 1 n = 8: F 2 8 = F 8 F 8+1 F 8 F 8 1 F 2 8 = F 8 F 9 F 8 F = = = 441 It works for n = 8!
38 Trying to prove: F F F F 2 n = F n F n+1 Let s use this formula: F 2 n = F n F n+1 F n F n 1 To find out what the sum of the first few SQUARES of Fibonacci numbers is: F F F F 2 n =????
39 Trying to prove: F F F F 2 n = F n F n+1 We re using: F 2 n = F n F n+1 F n F n 1 F1 2 = F 1 F 2 F2 2 = F 2 F 3 F 2 F 1 F3 2 = F 3 F 4 F 3 F 2 F4 2 = F 4 F 5 F 4 F 3 F 2 n 1 = F n 1 F n F n 1 F n 2 + F 2 n = F n F n+1 F n F n 1
40 Trying to prove: F F F F 2 n = F n F n+1 F1 2 = ///////// F 1 F 2 F2 2 = F 2 F 3 ///////// F 2 F 1 F 2 3 = F 3 F 4 F 3 F 2 F 2 4 = F 4 F 5 F 4 F 3 F 2 n 1 = F n 1 F n F n 1 F n 2 + F 2 n = F n F n+1 F n F n 1
41 Trying to prove: F F F F 2 n = F n F n+1 F1 2 = ///////// F 1 F 2 F2 2 = ///////// F 2 F 3 ///////// F 2 F 1 F3 2 = F 3 F 4 ///////// F 3 F 2 F 2 4 = F 4 F 5 F 4 F 3 F 2 n 1 = F n 1 F n F n 1 F n 2 + F 2 n = F n F n+1 F n F n 1
42 Trying to prove: F F F F 2 n = F n F n+1 F1 2 = ///////// F 1 F 2 F2 2 = ///////// F 2 F 3 ///////// F 2 F 1 F3 2 = ///////// F 3 F 4 ///////// F 3 F 2 F4 2 = F 4 F 5 ///////// F 4 F 3 F 2 n 1 = F n 1 F n F n 1 F n 2 + F 2 n = F n F n+1 F n F n 1
43 Trying to prove: F F F F 2 n = F n F n+1 F1 2 = ///////// F 1 F 2 F2 2 = ///////// F 2 F 3 ///////// F 2 F 1 F3 2 = ///////// F 3 F 4 ///////// F 3 F 2 F4 2 = ///////// F 4 F 5 ///////// F 4 F 3 Fn 1 2 = F n 1 F n F/////////////// n 1 F n 2 + F 2 n = F n F n+1 F n F n 1
44 Trying to prove: F F F F 2 n = F n F n+1 F1 2 = ///////// F 1 F 2 F2 2 = ///////// F 2 F 3 ///////// F 2 F 1 F3 2 = ///////// F 3 F 4 ///////// F 3 F 2 F4 2 = ///////// F 4 F 5 ///////// F 4 F 3 Fn 1 2 = F/////////////// n 1 F n+1 F/////////////// n 1 F n 2 + Fn 2 = F n F n+1 //////////// F n F n 1 F F F F 2 n = F n F n+1
45 Our fourth theorem: Theorem (Sum of first few SQUARES of Fibonacci numbers) For any positive integer n, the Fibonacci numbers satisfy: F1 2 + F2 2 + F Fn 2 = F n F n+1
46 So far: Theorem (Sum of first few Fibonacci numbers) F 1 + F 2 + F F n = F n+2 1 Theorem (Sum of first few EVEN Fibonacci numbers) F 2 + F 4 + F F 2n = F 2n+1 1 Theorem (Sum of first few ODD Fibonacci numbers) F 1 + F 3 + F F 2n 1 = F 2n Theorem (Sum of first few SQUARES of Fibonacci numbers) F F F F 2 n = F n F n+1
47 Examples = = = Now, complete this worksheet
Math 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 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 informationVieta s Formulas and the Identity Theorem
Vieta s Formulas and the Identity Theorem This worksheet will work through the material from our class on 3/21/2013 with some examples that should help you with the homework The topic of our discussion
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 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 informationSolution to Exercise 2.2. Both m and n are divisible by d, som = dk and n = dk. Thus m ± n = dk ± dk = d(k ± k ),som + n and m n are divisible by d.
[Chap. ] Pythagorean Triples 6 (b) The table suggests that in every primitive Pythagorean triple, exactly one of a, b,orc is a multiple of 5. To verify this, we use the Pythagorean Triples Theorem to write
More informationPigeonhole Principle Solutions
Pigeonhole Principle Solutions 1. Show that if we take n + 1 numbers from the set {1, 2,..., 2n}, then some pair of numbers will have no factors in common. Solution: Note that consecutive numbers (such
More informationFactorizations: Searching for Factor Strings
" 1 Factorizations: Searching for Factor Strings Some numbers can be written as the product of several different pairs of factors. For example, can be written as 1, 0,, 0, and. It is also possible to write
More informationZero: If P is a polynomial and if c is a number such that P (c) = 0 then c is a zero of P.
MATH 11011 FINDING REAL ZEROS KSU OF A POLYNOMIAL Definitions: Polynomial: is a function of the form P (x) = a n x n + a n 1 x n 1 + + a x + a 1 x + a 0. The numbers a n, a n 1,..., a 1, a 0 are called
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 information3. Mathematical Induction
3. MATHEMATICAL INDUCTION 83 3. Mathematical Induction 3.1. First Principle of Mathematical Induction. Let P (n) be a predicate with domain of discourse (over) the natural numbers N = {0, 1,,...}. If (1)
More informationSession 6 Number Theory
Key Terms in This Session Session 6 Number Theory Previously Introduced counting numbers factor factor tree prime number New in This Session composite number greatest common factor least common multiple
More informationWRITING PROOFS. Christopher Heil Georgia Institute of Technology
WRITING PROOFS Christopher Heil Georgia Institute of Technology A theorem is just a statement of fact A proof of the theorem is a logical explanation of why the theorem is true Many theorems have this
More informationQuestion: What is the probability that a five-card poker hand contains a flush, that is, five cards of the same suit?
ECS20 Discrete Mathematics Quarter: Spring 2007 Instructor: John Steinberger Assistant: Sophie Engle (prepared by Sophie Engle) Homework 8 Hints Due Wednesday June 6 th 2007 Section 6.1 #16 What is the
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 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 information1.3 Polynomials and Factoring
1.3 Polynomials and Factoring Polynomials Constant: a number, such as 5 or 27 Variable: a letter or symbol that represents a value. Term: a constant, variable, or the product or a constant and variable.
More informationProperties of Real Numbers
16 Chapter P Prerequisites P.2 Properties of Real Numbers What you should learn: Identify and use the basic properties of real numbers Develop and use additional properties of real numbers Why you should
More informationTAKE-AWAY GAMES. ALLEN J. SCHWENK California Institute of Technology, Pasadena, California INTRODUCTION
TAKE-AWAY GAMES ALLEN J. SCHWENK California Institute of Technology, Pasadena, California L INTRODUCTION Several games of Tf take-away?f have become popular. The purpose of this paper is to determine the
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 informationDiscrete 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
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 informationStanford 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
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 informationMathematical Induction. Lecture 10-11
Mathematical Induction Lecture 10-11 Menu Mathematical Induction Strong Induction Recursive Definitions Structural Induction Climbing an Infinite Ladder Suppose we have an infinite ladder: 1. We can reach
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 informationPOLYNOMIAL 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
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 informationHomework until Test #2
MATH31: Number Theory Homework until Test # Philipp BRAUN Section 3.1 page 43, 1. It has been conjectured that there are infinitely many primes of the form n. Exhibit five such primes. Solution. Five such
More information2.5 Zeros of a Polynomial Functions
.5 Zeros of a Polynomial Functions Section.5 Notes Page 1 The first rule we will talk about is Descartes Rule of Signs, which can be used to determine the possible times a graph crosses the x-axis and
More information2.2 Derivative as a Function
2.2 Derivative as a Function Recall that we defined the derivative as f (a) = lim h 0 f(a + h) f(a) h But since a is really just an arbitrary number that represents an x-value, why don t we just use x
More informationLecture 13 - Basic Number Theory.
Lecture 13 - Basic Number Theory. Boaz Barak March 22, 2010 Divisibility and primes Unless mentioned otherwise throughout this lecture all numbers are non-negative integers. We say that A divides B, denoted
More informationBasics of Counting. The product rule. Product rule example. 22C:19, Chapter 6 Hantao Zhang. Sample question. Total is 18 * 325 = 5850
Basics of Counting 22C:19, Chapter 6 Hantao Zhang 1 The product rule Also called the multiplication rule If there are n 1 ways to do task 1, and n 2 ways to do task 2 Then there are n 1 n 2 ways to do
More informationMATH 289 PROBLEM SET 4: NUMBER THEORY
MATH 289 PROBLEM SET 4: NUMBER THEORY 1. The greatest common divisor If d and n are integers, then we say that d divides n if and only if there exists an integer q such that n = qd. Notice that if d divides
More informationThe Pointless Machine and Escape of the Clones
MATH 64091 Jenya Soprunova, KSU The Pointless Machine and Escape of the Clones The Pointless Machine that operates on ordered pairs of positive integers (a, b) has three modes: In Mode 1 the machine adds
More information0.8 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
More informationIB Math Research Problem
Vincent Chu Block F IB Math Research Problem The product of all factors of 2000 can be found using several methods. One of the methods I employed in the beginning is a primitive one I wrote a computer
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 information3.3 Real Zeros of Polynomials
3.3 Real Zeros of Polynomials 69 3.3 Real Zeros of Polynomials In Section 3., we found that we can use synthetic division to determine if a given real number is a zero of a polynomial function. This section
More informationSample Induction Proofs
Math 3 Worksheet: Induction Proofs III, Sample Proofs A.J. Hildebrand Sample Induction Proofs Below are model solutions to some of the practice problems on the induction worksheets. The solutions given
More informationGuide to Leaving Certificate Mathematics Ordinary Level
Guide to Leaving Certificate Mathematics Ordinary Level Dr. Aoife Jones Paper 1 For the Leaving Cert 013, Paper 1 is divided into three sections. Section A is entitled Concepts and Skills and contains
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 informationZeros 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
More informationSection 6.4: Counting Subsets of a Set: Combinations
Section 6.4: Counting Subsets of a Set: Combinations In section 6.2, we learnt how to count the number of r-permutations from an n-element set (recall that an r-permutation is an ordered selection of r
More informationSECTION 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
More informationPage 331, 38.4 Suppose a is a positive integer and p is a prime. Prove that p a if and only if the prime factorization of a contains p.
Page 331, 38.2 Assignment #11 Solutions Factor the following positive integers into primes. a. 25 = 5 2. b. 4200 = 2 3 3 5 2 7. c. 10 10 = 2 10 5 10. d. 19 = 19. e. 1 = 1. Page 331, 38.4 Suppose a is a
More informations = 1 + 2 +... + 49 + 50 s = 50 + 49 +... + 2 + 1 2s = 51 + 51 +... + 51 + 51 50 51. 2
1. Use Euler s trick to find the sum 1 + 2 + 3 + 4 + + 49 + 50. s = 1 + 2 +... + 49 + 50 s = 50 + 49 +... + 2 + 1 2s = 51 + 51 +... + 51 + 51 Thus, 2s = 50 51. Therefore, s = 50 51. 2 2. Consider the sequence
More informationCS311H. Prof: Peter Stone. Department of Computer Science The University of Texas at Austin
CS311H Prof: Department of Computer Science The University of Texas at Austin Good Morning, Colleagues Good Morning, Colleagues Are there any questions? Logistics Class survey Logistics Class survey Homework
More informationSolving Quadratic Equations
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
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 informationLS.6 Solution Matrices
LS.6 Solution Matrices In the literature, solutions to linear systems often are expressed using square matrices rather than vectors. You need to get used to the terminology. As before, we state the definitions
More informationPrime Time: Homework Examples from ACE
Prime Time: Homework Examples from ACE Investigation 1: Building on Factors and Multiples, ACE #8, 28 Investigation 2: Common Multiples and Common Factors, ACE #11, 16, 17, 28 Investigation 3: Factorizations:
More informationMathematics Success Grade 6
T276 Mathematics Success Grade 6 [OBJECTIVE] The student will add and subtract with decimals to the thousandths place in mathematical and real-world situations. [PREREQUISITE SKILLS] addition and subtraction
More informationMathematical Induction. Mary Barnes Sue Gordon
Mathematics Learning Centre Mathematical Induction Mary Barnes Sue Gordon c 1987 University of Sydney Contents 1 Mathematical Induction 1 1.1 Why do we need proof by induction?.... 1 1. What is proof by
More informationBasic Proof Techniques
Basic Proof Techniques David Ferry dsf43@truman.edu September 13, 010 1 Four Fundamental Proof Techniques When one wishes to prove the statement P Q there are four fundamental approaches. This document
More information26 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
More informationComputing exponents modulo a number: Repeated squaring
Computing exponents modulo a number: Repeated squaring How do you compute (1415) 13 mod 2537 = 2182 using just a calculator? Or how do you check that 2 340 mod 341 = 1? You can do this using the method
More informationSection IV.1: Recursive Algorithms and Recursion Trees
Section IV.1: Recursive Algorithms and Recursion Trees Definition IV.1.1: A recursive algorithm is an algorithm that solves a problem by (1) reducing it to an instance of the same problem with smaller
More informationRevised Version of Chapter 23. We learned long ago how to solve linear congruences. ax c (mod m)
Chapter 23 Squares Modulo p Revised Version of Chapter 23 We learned long ago how to solve linear congruences ax c (mod m) (see Chapter 8). It s now time to take the plunge and move on to quadratic equations.
More informationSession 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
More informationMATHEMATICAL INDUCTION. Mathematical Induction. This is a powerful method to prove properties of positive integers.
MATHEMATICAL INDUCTION MIGUEL A LERMA (Last updated: February 8, 003) Mathematical Induction This is a powerful method to prove properties of positive integers Principle of Mathematical Induction Let P
More informationTeaching & 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.
More informationThe Function Game: Can You Guess the Secret?
The Function Game: Can You Guess the Secret? Copy the input and output numbers for each secret given by your teacher. Write your guess for what is happening to the input number to create the output number
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 informationRecursive Algorithms. Recursion. Motivating Example Factorial Recall the factorial function. { 1 if n = 1 n! = n (n 1)! if n > 1
Recursion Slides by Christopher M Bourke Instructor: Berthe Y Choueiry Fall 007 Computer Science & Engineering 35 Introduction to Discrete Mathematics Sections 71-7 of Rosen cse35@cseunledu Recursive Algorithms
More informationLecture Notes on Polynomials
Lecture Notes on Polynomials Arne Jensen Department of Mathematical Sciences Aalborg University c 008 Introduction These lecture notes give a very short introduction to polynomials with real and complex
More information3.1. 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
More informationLesson 9: Radicals and Conjugates
Student Outcomes Students understand that the sum of two square roots (or two cube roots) is not equal to the square root (or cube root) of their sum. Students convert expressions to simplest radical form.
More informationLaw of Cosines. If the included angle is a right angle then the Law of Cosines is the same as the Pythagorean Theorem.
Law of Cosines In the previous section, we learned how the Law of Sines could be used to solve oblique triangles in three different situations () where a side and two angles (SAA) were known, () where
More informationDecimal Notations for Fractions Number and Operations Fractions /4.NF
Decimal Notations for Fractions Number and Operations Fractions /4.NF Domain: Cluster: Standard: 4.NF Number and Operations Fractions Understand decimal notation for fractions, and compare decimal fractions.
More informationApplications of Fermat s Little Theorem and Congruences
Applications of Fermat s Little Theorem and Congruences Definition: Let m be a positive integer. Then integers a and b are congruent modulo m, denoted by a b mod m, if m (a b). Example: 3 1 mod 2, 6 4
More informationMultiplication and Division with Rational Numbers
Multiplication and Division with Rational Numbers Kitty Hawk, North Carolina, is famous for being the place where the first airplane flight took place. The brothers who flew these first flights grew up
More informationSolution to Homework 2
Solution to Homework 2 Olena Bormashenko September 23, 2011 Section 1.4: 1(a)(b)(i)(k), 4, 5, 14; Section 1.5: 1(a)(b)(c)(d)(e)(n), 2(a)(c), 13, 16, 17, 18, 27 Section 1.4 1. Compute the following, if
More informationparent ROADMAP MATHEMATICS SUPPORTING YOUR CHILD IN HIGH SCHOOL
parent ROADMAP MATHEMATICS SUPPORTING YOUR CHILD IN HIGH SCHOOL HS America s schools are working to provide higher quality instruction than ever before. The way we taught students in the past simply does
More informationUmmmm! Definitely interested. She took the pen and pad out of my hand and constructed a third one for herself:
Sum of Cubes Jo was supposed to be studying for her grade 12 physics test, but her soul was wandering. Show me something fun, she said. Well I wasn t sure just what she had in mind, but it happened that
More informationGEOMETRIC 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
More informationPearson Algebra 1 Common Core 2015
A Correlation of Pearson Algebra 1 Common Core 2015 To the Common Core State Standards for Mathematics Traditional Pathways, Algebra 1 High School Copyright 2015 Pearson Education, Inc. or its affiliate(s).
More informationFRACTIONS OPERATIONS
FRACTIONS OPERATIONS Summary 1. Elements of a fraction... 1. Equivalent fractions... 1. Simplification of a fraction... 4. Rules for adding and subtracting fractions... 5. Multiplication rule for two fractions...
More informationAcquisition Lesson Planning Form Key Standards addressed in this Lesson: MM2A3d,e Time allotted for this Lesson: 4 Hours
Acquisition Lesson Planning Form Key Standards addressed in this Lesson: MM2A3d,e Time allotted for this Lesson: 4 Hours Essential Question: LESSON 4 FINITE ARITHMETIC SERIES AND RELATIONSHIP TO QUADRATIC
More informationMath Circle Beginners Group October 18, 2015
Math Circle Beginners Group October 18, 2015 Warm-up problem 1. Let n be a (positive) integer. Prove that if n 2 is odd, then n is also odd. (Hint: Use a proof by contradiction.) Suppose that n 2 is odd
More informationArithmetic 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
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 3. Methods of Proofs. 1. Logical Arguments and Formal Proofs
CHAPTER 3 Methods of Proofs 1. Logical Arguments and Formal Proofs 1.1. Basic Terminology. An axiom is a statement that is given to be true. A rule of inference is a logical rule that is used to deduce
More informationWarm up. Connect these nine dots with only four straight lines without lifting your pencil from the paper.
Warm up Connect these nine dots with only four straight lines without lifting your pencil from the paper. Sometimes we need to think outside the box! Warm up Solution Warm up Insert the Numbers 1 8 into
More informationCongruent Number Problem
University of Waterloo October 28th, 2015 Number Theory Number theory, can be described as the mathematics of discovering and explaining patterns in numbers. There is nothing in the world which pleases
More informationMath 319 Problem Set #3 Solution 21 February 2002
Math 319 Problem Set #3 Solution 21 February 2002 1. ( 2.1, problem 15) Find integers a 1, a 2, a 3, a 4, a 5 such that every integer x satisfies at least one of the congruences x a 1 (mod 2), x a 2 (mod
More informationPythagorean Triples. becomes
Solution Commentary: Solution of Main Problems: Pythagorean Triples 1. If m = (n + 1), n = [m -1]/ and n+1 = [m +1]/. Then, by substitution, the equation n + (n + 1) = (n+1) m + 1 becomes + m =. Now, because
More informationRECURSIVE ENUMERATION OF PYTHAGOREAN TRIPLES
RECURSIVE ENUMERATION OF PYTHAGOREAN TRIPLES DARRYL MCCULLOUGH AND ELIZABETH WADE In [9], P. W. Wade and W. R. Wade (no relation to the second author gave a recursion formula that produces Pythagorean
More informationALGEBRA. 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,
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 informationDiscrete Mathematics: Homework 7 solution. Due: 2011.6.03
EE 2060 Discrete Mathematics spring 2011 Discrete Mathematics: Homework 7 solution Due: 2011.6.03 1. Let a n = 2 n + 5 3 n for n = 0, 1, 2,... (a) (2%) Find a 0, a 1, a 2, a 3 and a 4. (b) (2%) Show that
More information7 th Grade Integer Arithmetic 7-Day Unit Plan by Brian M. Fischer Lackawanna Middle/High School
7 th Grade Integer Arithmetic 7-Day Unit Plan by Brian M. Fischer Lackawanna Middle/High School Page 1 of 20 Table of Contents Unit Objectives........ 3 NCTM Standards.... 3 NYS Standards....3 Resources
More informationThe Fibonacci Sequence and the Golden Ratio
55 The solution of Fibonacci s rabbit problem is examined in Chapter, pages The Fibonacci Sequence and the Golden Ratio The Fibonacci Sequence One of the most famous problems in elementary mathematics
More informationMathematical Induction
Mathematical Induction (Handout March 8, 01) The Principle of Mathematical Induction provides a means to prove infinitely many statements all at once The principle is logical rather than strictly mathematical,
More informationNumber Theory. Proof. Suppose otherwise. Then there would be a finite number n of primes, which we may
Number Theory Divisibility and Primes Definition. If a and b are integers and there is some integer c such that a = b c, then we say that b divides a or is a factor or divisor of a and write b a. Definition
More information2 When is a 2-Digit Number the Sum of the Squares of its Digits?
When Does a Number Equal the Sum of the Squares or Cubes of its Digits? An Exposition and a Call for a More elegant Proof 1 Introduction We will look at theorems of the following form: by William Gasarch
More informationON FIBONACCI NUMBERS WITH FEW PRIME DIVISORS
ON FIBONACCI NUMBERS WITH FEW PRIME DIVISORS YANN BUGEAUD, FLORIAN LUCA, MAURICE MIGNOTTE, SAMIR SIKSEK Abstract If n is a positive integer, write F n for the nth Fibonacci number, and ω(n) for the number
More information2.1. Inductive Reasoning EXAMPLE A
CONDENSED LESSON 2.1 Inductive Reasoning In this lesson you will Learn how inductive reasoning is used in science and mathematics Use inductive reasoning to make conjectures about sequences of numbers
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 informationJacobi s four squares identity Martin Klazar
Jacobi s four squares identity Martin Klazar (lecture on the 7-th PhD conference) Ostrava, September 10, 013 C. Jacobi [] in 189 proved that for any integer n 1, r (n) = #{(x 1, x, x 3, x ) Z ( i=1 x i
More information