Math 115 Spring 2011 Written Homework 5 Solutions


 Jerome McDonald
 2 years ago
 Views:
Transcription
1 . Evaluate each series. a) 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 is arithmetic, then the common difference is d = 7 4 = 0 7 = 3. Then the candidate for the generating function is an) := 4 n )3. To prove that the associated sequence is arithmetic, we need to show that 55 is a term generated by the function an). That is, there is some N where an) = 55. an) = 55 4 N )3 = 55 3N = 55 3N = 54 N = 8 Since N = 8 is a natural number, the series is an arithmetic series. Using the summation formula for a finite arithmetic series, ) ) a a N 4 55 S = N = 8 = 959) = 53. b) The first and last terms of summation are 8 and, respectively and the common ratio 5 between each term is 4. Solution: Here we are given that the series ) is a geometric series. The generating function n for the associated sequence is bn) := 8. For a geometric series, we have two forms 4 of the summation formula, S = a N a r ) r N and S = a where N is the number of r
2 terms in the series. While we could find N using the generating function, since we know the first and last terms in the series, we don t need to. Here b = 8 and b N = 5. S = b N b r = 4 3 = b N)r b r = 5 ) 4 ) 8 4 ) = = ) = ) = = ) c) The sum of where the associated sequence has terms. Solution: We note that the associated sequence, 3, 6,, 4,... is a geometric sequence. r = 6 3 = 6 = 4 = The generating function for the sequence is cn) := 3) ) n. We are given that the series has terms. Here we use the other formulation for the sum of a finite geometric series. ) ) r N ) S = a = 3) = ). r ) d) The sum of 0, 5, 5, 5,... where the associated sequence has 30 terms. Solution: This is an arithmetic sequence: d = = 5 = 5 5 = 5. Then dn) := 0 n ) 5 ) is the generating function for the associated sequence. We are given that N = 30. Then ) ) )) a a 30 0 [0 30 ) 5 S = 30 = 30 ) ] = ) = 5 05 ) = 575.
3 . How many terms of the sequence 5,, 3,... must be added to give a sum of 400? Solution: We need to first determine if this sequence is arithmetic or geometric. Since 5) = 3 ) = 4, we assume that the sequence is arithmetic with a common difference d = 4. Then, the generating function of the sequence is an) := 5 n )4). The summation formula for the associated Nterm arithmetic series is ) ) a a N 5 5 N )4) S = N = N. We are given that the summation S = 400. We need to determine N. ) 5 5 N )4) 400 = N ) 0 4N 4 = N ) 4 4N = N = N 7 N) = 7N N 0 = N 7N 400 Factor the above quadratic equation or use the quadratic formula to solve for N. N = 7) ± 7) 4) 400). ) N = 7 ± 57 = 6 or 5. Now, both of these candidates for N can not be correct. Recall that N is a number of terms in a sequence / series. N must be a natural number. Thus, N = 6 is the only solution.
4 3. a) Use a series to find the sum of the first 00 odd, positive integers. Solution: We know from lecture that the sequence that generates the odd integer is oddn) := n ) = n. The series here is odd00) = [ 00)] = The summation formula for this arithmetic series is ) 399 S = 00 = b) Use a series to find the sum of all positive integers less than 00 that are multiples of 7. Solution: The series here is N where N is the largest natural number where 7N 00. Hence, N = 8. N 00 7 = We notice that the associated sequence is arithmetic: bn) := 7n. The summation formula for this arithmetic series is ) 7 78) S = 8 = 40) = 84.
5 4. How many terms of the sequence generated by the function a n := 43) n must be added to give a sum of 456? Solution: The associated series is clearly geometric. The summation formula for an N term geometric series is S = a r N r ) ) 3 N 456 = = 3N 78 = 3 N 79 = 3 N 79 = 3 N 3 6 = 3 N N = 6 There are 6 terms in the associated series. 5. If 0 a, 0 a, 0 a 3,..., 0 an is a geometric sequence, what can you determine about the sequence a, a, a 3,..., a n? Solution: We are given that 0 a, 0 a, 0 a 3,..., 0 an is a geometric sequence. Thus, r = 0a 0 a = 0a3 0 a = 0a4 0 a 3 =... = 0an 0 a n = 0 a a = 0 a 3 a = 0 a 4 a 3 =... = 0 an a n For the numbers to all be equal, the exponent on the base 0 must be the same. That is, there is some exponent p where p = a a = a 3 a = a 4 a 3 =... = a n a n. The equation a n a n = p for all n is the definition of an arithmetic sequence with common difference p. Hence, the sequence a, a, a 3,..., a n must be an arithmetic sequence.
6 9 6. Write π k as an expanded sum and compute the sum. k=3 9 Solution: π k = π 3 π 4 π 5 π 6 π 7 π 8 π 9. k=3 Recall that if a number x is negative then x = x. Additionally, recall that π Hence 6 < π < 7. Thus, π k = π k when k 6 and π k = π k), when k > 6. Then, 9 π k = π 3) π 4) π 5) π 6) [ π 7)] [ π 8)] [ π 9)] k=3 = π 3 π 4 π 5 π 6 π 7) π 8) π 9) = π 3 π 4 π 5 π 6 π 7 π 8 π 9 = π = π 6
7 7. Write each of the following using summation notation. a) a b ) a b ) 3 a 3 b 3 ) 4... a 0 b 0 ) 0 Solution: a i b i ) i. i= b) The sum of all three digit positive even integers. n= Solution: This is the series This associated sequence is arithmetic and is generated by the function bn) := 00 n ). Then the summation is N bn). We need to know how many terms are in the series in order to define the upperbound on the index in our summation notation. Note that we use bn) = 998 to determine how many terms are in the sequence. 998 = 00 N ) 898 = N ) 449 = N N = Then the series can be written in the form [00 n )]. n= Remark: An alternative method would result in the answer c) n=50 n. This is also correct. Solution: Again, in order to write this as a summation, we need a generating function for the associate sequence. We are lead to believe that the associate sequence is geometric because r = 6 = /3 = /9 /3 = 3. To be certain that this series is a geometric series, we need to show that the term 43 is a term in the sequence generated by cn) := 6 n. If it is, in this process we will 3)
8 determine how many numbers are in this sequence.) cn) = 43 6 ) N = 3 43 ) N = 3 79 N = 3) 3 N = 6 N = 7 ) 6 Thus, = 6 ) n 3 n=
9 4 8. If a b ab) = b= 5 ac 6), determine a. c=3 Solution: Here we expand and simplify. 4 a b ab) = b= 5 ac 6) c=3 a a) a 3 a3) a 4 a4) = a3 6) a4 6) a5 6) a a 3a 3a 4a 4a = 3a 6 4a 6 5a 6 9a 9a = a 8 9a a 8 = 0 3a 7a 6) = 0 3a 3)3a ) = 0 Here, the summations will be equal when a is either a = 3 or a =. 3
10 n 9. What is the sum of the series ) k, if n is odd? if n is even? Solution: Try some values for n. ) = ) = ) ) = = 0 3 ) = ) ) ) 3 = = 4 ) = ) ) ) 3 ) 4 = = 0 5 ) = ) ) ) 3 ) 4 ) 5 = = At this point, we recognize the pattern. If n is an even number, we can form n/ pairs of = 0 and the summation will always be 0. If n is an odd number, the pairs made by the first n terms will cancel and we will be left with a single. Hence, the summation equals  when n is odd.
11 0. a) Write 0 kk ) as an expanded sum and compute the sum. Solution: 0 kk ) = ) 3) 34) 45) 56) 67) 78) 89) 90) 0) = = 0 0 b) The summation k ) is an example of a telescoping sum. Expand and k compute this sum. What property of the summation makes this a telescoping sum? Solution: 0 k ) k = ) ) 3 3 ) 4 4 ) 5 5 ) 6 6 ) 7 7 ) 8 8 ) 9 9 ) 0 0 ) = = = 0 The property that makes this a telescoping sum is the fact that it collapses down to a much smaller sum just as a telescope can expand and collapse). c) Let N be a large positive number. Evaluate N k ). k Solution: N k ) k = ) ) 3 3 )... 4 N ) N = N or N N
12 d) Show that k k = kk ). Solution: e) Solution: Evaluate k k = k kk ) k kk ) = k k kk ) = kk ) 000 kk ) kk ) = k ) = k 00 =
Math 115 Spring 2014 Written Homework 3 Due Wednesday, February 19
Math 11 Spring 01 Written Homework 3 Due Wednesday, February 19 Instructions: Write complete solutions on separate paper (not spiral bound). If multiple pieces of paper are used, they must be stapled with
More information8.7 Mathematical Induction
8.7. MATHEMATICAL INDUCTION 8135 8.7 Mathematical Induction Objective Prove a statement by mathematical induction Many mathematical facts are established by first observing a pattern, then making a conjecture
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 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 information16.1. Sequences and Series. Introduction. Prerequisites. Learning Outcomes. Learning Style
Sequences and Series 16.1 Introduction In this block we develop the ground work for later blocks on infinite series and on power series. We begin with simple sequences of numbers and with finite series
More information8.3. GEOMETRIC SEQUENCES AND SERIES
8.3. GEOMETRIC SEQUENCES AND SERIES What You Should Learn Recognize, write, and find the nth terms of geometric sequences. Find the sum of a finite geometric sequence. Find the sum of an infinite geometric
More informationTransition to College Math Period
Transition to College Math Name Period Date: Unit: 1: Series and Sequences Lesson: 3: Geometric Sequences Standard: F.LE.2 Learning Target: Geometric Sequence: Essential Question: The average of two numbers
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 informationTo 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
More informationAppendix F: Mathematical Induction
Appendix F: Mathematical Induction Introduction In this appendix, you will study a form of mathematical proof called mathematical induction. To see the logical need for mathematical induction, take another
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 informationExam #5 Sequences and Series
College of the Redwoods Mathematics Department Math 30 College Algebra Exam #5 Sequences and Series David Arnold Don Hickethier Copyright c 000 DonHickethier@Eureka.redwoods.cc.ca.us Last Revision Date:
More informationSequences and Series
Contents 6 Sequences and Series 6. Sequences and Series 6. Infinite Series 3 6.3 The Binomial Series 6 6.4 Power Series 3 6.5 Maclaurin and Taylor Series 40 Learning outcomes In this Workbook you will
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 information2.4 Multiplication of Integers. Recall that multiplication is defined as repeated addition from elementary school. For example, 5 6 = 6 5 = 30, since:
2.4 Multiplication of Integers Recall that multiplication is defined as repeated addition from elementary school. For example, 5 6 = 6 5 = 30, since: 5 6=6+6+6+6+6=30 6 5=5+5+5+5+5+5=30 To develop a rule
More information1. The algebra of exponents 1.1. Natural Number Powers. It is easy to say what is meant by a n a (raised to) to the (power) n if n N.
CHAPTER 3: EXPONENTS AND POWER FUNCTIONS 1. The algebra of exponents 1.1. Natural Number Powers. It is easy to say what is meant by a n a (raised to) to the (power) n if n N. For example: In general, if
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 informationMathematical Induction
Chapter 2 Mathematical Induction 2.1 First Examples Suppose we want to find a simple formula for the sum of the first n odd numbers: 1 + 3 + 5 +... + (2n 1) = n (2k 1). How might we proceed? The most natural
More information6 Rational Inequalities, (In)equalities with Absolute value; Exponents and Logarithms
AAU  Business Mathematics I Lecture #6, March 16, 2009 6 Rational Inequalities, (In)equalities with Absolute value; Exponents and Logarithms 6.1 Rational Inequalities: x + 1 x 3 > 1, x + 1 x 2 3x + 5
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 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 information#112: Write the first 4 terms of the sequence. (Assume n begins with 1.)
Section 9.1: Sequences #112: Write the first 4 terms of the sequence. (Assume n begins with 1.) 1) a n = 3n a 1 = 3*1 = 3 a 2 = 3*2 = 6 a 3 = 3*3 = 9 a 4 = 3*4 = 12 3) a n = 3n 5 Answer: 3,6,9,12 a 1
More informationSECTION 102 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 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 information10.2 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
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 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 informationSequences and Mathematical Induction. CSE 215, Foundations of Computer Science Stony Brook University
Sequences and Mathematical Induction CSE 215, Foundations of Computer Science Stony Brook University http://www.cs.stonybrook.edu/~cse215 Sequences A sequence is a function whose domain is all the integers
More informationEquations and Inequalities
Rational Equations Overview of Objectives, students should be able to: 1. Solve rational equations with variables in the denominators.. Recognize identities, conditional equations, and inconsistent equations.
More informationLecture 3. Mathematical Induction
Lecture 3 Mathematical Induction Induction is a fundamental reasoning process in which general conclusion is based on particular cases It contrasts with deduction, the reasoning process in which conclusion
More informationCS 173: Discrete Structures, Fall 2010 Homework 6 Solutions
CS 173: Discrete Structures, Fall 010 Homework 6 Solutions This homework was worth a total of 5 points. 1. Recursive definition [13 points] Give a simple closedform definition for each of the following
More informationCSI 333 Lecture 1 Number Systems
CSI 333 Lecture 1 Number Systems 1 1 / 23 Basics of Number Systems Ref: Appendix C of Deitel & Deitel. Weighted Positional Notation: 192 = 2 10 0 + 9 10 1 + 1 10 2 General: Digit sequence : d n 1 d n 2...
More informationSan Jose Math Circle October 17, 2009 ARITHMETIC AND GEOMETRIC PROGRESSIONS
San Jose Math Circle October 17, 2009 ARITHMETIC AND GEOMETRIC PROGRESSIONS DEFINITION. An arithmetic progression is a (finite or infinite) sequence of numbers with the property that the difference between
More informationSometimes it is easier to leave a number written as an exponent. For example, it is much easier to write
4.0 Exponent Property Review First let s start with a review of what exponents are. Recall that 3 means taking four 3 s and multiplying them together. So we know that 3 3 3 3 381. You might also recall
More informationSucesiones repaso Klasse 12 [86 marks]
Sucesiones repaso Klasse [8 marks] a. Consider an infinite geometric sequence with u = 40 and r =. (i) Find. u 4 (ii) Find the sum of the infinite sequence. (i) correct approach () e.g. u 4 = (40) u 4
More informationIB Math 11 Assignment: Chapters 1 & 2 (A) NAME: (Functions, Sequences and Series)
IB Math 11 Assignment: Chapters 1 & 2 (A) NAME: (Functions, Sequences and Series) 1. Let f(x) = 7 2x and g(x) = x + 3. Find (g f)(x). Write down g 1 (x). (c) Find (f g 1 )(5). (Total 5 marks) 2. Consider
More information81 Adding and Subtracting Polynomials
Determine whether each expression is a polynomial. If it is a polynomial, find the degree and determine whether it is a monomial, binomial, or trinomial. 1. 7ab + 6b 2 2a 3 yes; 3; trinomial 2. 2y 5 +
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 informationGrade Level Year Total Points Core Points % At Standard 9 2003 10 5 7 %
Performance Assessment Task Number Towers Grade 9 The task challenges a student to demonstrate understanding of the concepts of algebraic properties and representations. A student must make sense of the
More informationSequences with MS
Sequences 20082014 with MS 1. [4 marks] Find the value of k if. 2a. [4 marks] The sum of the first 16 terms of an arithmetic sequence is 212 and the fifth term is 8. Find the first term and the common
More informationLecture 5 Principal Minors and the Hessian
Lecture 5 Principal Minors and the Hessian Eivind Eriksen BI Norwegian School of Management Department of Economics October 01, 2010 Eivind Eriksen (BI Dept of Economics) Lecture 5 Principal Minors and
More informationChapter 4  Decimals
Chapter 4  Decimals $34.99 decimal notation ex. The cost of an object. ex. The balance of your bank account ex The amount owed ex. The tax on a purchase. Just like Whole Numbers Place Value  1.23456789
More informationa 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
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 informationCalculus for Middle School Teachers. Problems and Notes for MTHT 466
Calculus for Middle School Teachers Problems and Notes for MTHT 466 Bonnie Saunders Fall 2010 1 I Infinity Week 1 How big is Infinity? Problem of the Week: The Chess Board Problem There once was a humble
More informationExponent Properties Involving Products
Exponent Properties Involving Products Learning Objectives Use the product of a power property. Use the power of a product property. Simplify expressions involving product properties of exponents. Introduction
More informationPYTHAGOREAN TRIPLES KEITH CONRAD
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
More informationSection 8.4  Composite and Inverse Functions
Math 127  Section 8.4  Page 1 Section 8.4  Composite and Inverse Functions I. Composition of Functions A. If f and g are functions, then the composite function of f and g (written f g) is: (f g)( =
More information61. REARRANGEMENTS 119
61. REARRANGEMENTS 119 61. Rearrangements Here the difference between conditionally and absolutely convergent series is further refined through the concept of rearrangement. Definition 15. (Rearrangement)
More informationp 2 1 (mod 6) Adding 2 to both sides gives p (mod 6)
.9. Problems P10 Try small prime numbers first. p p + 6 3 11 5 7 7 51 11 13 Among the primes in this table, only the prime 3 has the property that (p + ) is also a prime. We try to prove that no other
More informationHomework 5 Solutions
Homework 5 Solutions 4.2: 2: a. 321 = 256 + 64 + 1 = (01000001) 2 b. 1023 = 512 + 256 + 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1 = (1111111111) 2. Note that this is 1 less than the next power of 2, 1024, which
More informationChapter 7  Roots, Radicals, and Complex Numbers
Math 233  Spring 2009 Chapter 7  Roots, Radicals, and Complex Numbers 7.1 Roots and Radicals 7.1.1 Notation and Terminology In the expression x the is called the radical sign. The expression under the
More informationA fairly quick tempo of solutions discussions can be kept during the arithmetic problems.
Distributivity and related number tricks Notes: No calculators are to be used Each group of exercises is preceded by a short discussion of the concepts involved and one or two examples to be worked out
More information1, 1 2, 1 3, 1 4,... 2 nd term. 1 st term
1 Sequences 11 Overview A (numerical) sequence is a list of real numbers in which each entry is a function of its position in the list The entries in the list are called terms For example, 1, 1, 1 3, 1
More informationCHAPTER 5: MODULAR ARITHMETIC
CHAPTER 5: MODULAR ARITHMETIC LECTURE NOTES FOR MATH 378 (CSUSM, SPRING 2009). WAYNE AITKEN 1. Introduction In this chapter we will consider congruence modulo m, and explore the associated arithmetic called
More informationMath 2602 Finite and Linear Math Fall 14. Homework 9: Core solutions
Math 2602 Finite and Linear Math Fall 14 Homework 9: Core solutions Section 8.2 on page 264 problems 13b, 27a27b. Section 8.3 on page 275 problems 1b, 8, 10a10b, 14. Section 8.4 on page 279 problems
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 informationOverview. 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
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 informationMATH 289 PROBLEM SET 1: INDUCTION. 1. The induction Principle The following property of the natural numbers is intuitively clear:
MATH 89 PROBLEM SET : INDUCTION The induction Principle The following property of the natural numbers is intuitively clear: Axiom Every nonempty subset of the set of nonnegative integers Z 0 = {0,,, 3,
More informationRepton Manor Primary School. Maths Targets
Repton Manor Primary School Maths Targets Which target is for my child? Every child at Repton Manor Primary School will have a Maths Target, which they will keep in their Maths Book. The teachers work
More informationAlgebra Tiles Activity 1: Adding Integers
Algebra Tiles Activity 1: Adding Integers NY Standards: 7/8.PS.6,7; 7/8.CN.1; 7/8.R.1; 7.N.13 We are going to use positive (yellow) and negative (red) tiles to discover the rules for adding and subtracting
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 informationQuadratic Equations and Inequalities
MA 134 Lecture Notes August 20, 2012 Introduction The purpose of this lecture is to... Introduction The purpose of this lecture is to... Learn about different types of equations Introduction The purpose
More informationAdvanced Engineering Mathematics Prof. Jitendra Kumar Department of Mathematics Indian Institute of Technology, Kharagpur
Advanced Engineering Mathematics Prof. Jitendra Kumar Department of Mathematics Indian Institute of Technology, Kharagpur Lecture No. # 28 Fourier Series (Contd.) Welcome back to the lecture on Fourier
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 informationFINAL EXAM SECTIONS AND OBJECTIVES FOR COLLEGE ALGEBRA
FINAL EXAM SECTIONS AND OBJECTIVES FOR COLLEGE ALGEBRA 1.1 Solve linear equations and equations that lead to linear equations. a) Solve the equation: 1 (x + 5) 4 = 1 (2x 1) 2 3 b) Solve the equation: 3x
More informationAnswer 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.
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 informationThe Deadly Sins of Algebra
The Deadly Sins of Algebra There are some algebraic misconceptions that are so damaging to your quantitative and formal reasoning ability, you might as well be said not to have any such reasoning ability.
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 information9.2 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
More informationWe can express this in decimal notation (in contrast to the underline notation we have been using) as follows: 9081 + 900b + 90c = 9001 + 100c + 10b
In this session, we ll learn how to solve problems related to place value. This is one of the fundamental concepts in arithmetic, something every elementary and middle school mathematics teacher should
More informationSection 9.5: Equations of Lines and Planes
Lines in 3D Space Section 9.5: Equations of Lines and Planes Practice HW from Stewart Textbook (not to hand in) p. 673 # 35 odd, 237 odd, 4, 47 Consider the line L through the point P = ( x, y, ) that
More informationMathematics 31 Precalculus and Limits
Mathematics 31 Precalculus and Limits Overview After completing this section, students will be epected to have acquired reliability and fluency in the algebraic skills of factoring, operations with radicals
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 information12.3 Geometric Sequences. Copyright Cengage Learning. All rights reserved.
12.3 Geometric Sequences Copyright Cengage Learning. All rights reserved. 1 Objectives Geometric Sequences Partial Sums of Geometric Sequences What Is an Infinite Series? Infinite Geometric Series 2 Geometric
More informationLinear, Quadratic, and Exponential Models
Linear, Quadratic, and Eponential Models Goal: To guide students to an understanding of the inverse relationship between logarithms and eponential functions, and of how to solve eponential equations and
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 informationSMT 2014 Algebra Test Solutions February 15, 2014
1. Alice and Bob are painting a house. If Alice and Bob do not take any breaks, they will finish painting the house in 20 hours. If, however, Bob stops painting once the house is halffinished, then the
More informationMath Review Large Print (18 point) Edition Chapter 2: Algebra
GRADUATE RECORD EXAMINATIONS Math Review Large Print (18 point) Edition Chapter : Algebra Copyright 010 by Educational Testing Service. All rights reserved. ETS, the ETS logo, GRADUATE RECORD EXAMINATIONS,
More informationSTUDY GUIDE FOR SOME BASIC INTERMEDIATE ALGEBRA SKILLS
STUDY GUIDE FOR SOME BASIC INTERMEDIATE ALGEBRA SKILLS The intermediate algebra skills illustrated here will be used extensively and regularly throughout the semester Thus, mastering these skills is an
More informationClick on the links below to jump directly to the relevant section
Click on the links below to jump directly to the relevant section Basic review Writing fractions in simplest form Comparing fractions Converting between Improper fractions and whole/mixed numbers Operations
More informationNotes: Geometric Sequences
Notes: Geometric Sequences I. What is a Geometric Sequence? The table shows the heights of a bungee jumper s bounces. The height of the bounces shown in the table above form a geometric sequence. In a
More informationMathematical induction. Niloufar Shafiei
Mathematical induction Niloufar Shafiei Mathematical induction Mathematical induction is an extremely important proof technique. Mathematical induction can be used to prove results about complexity of
More informationAccentuate the Negative: Homework Examples from ACE
Accentuate the Negative: Homework Examples from ACE Investigation 1: Extending the Number System, ACE #6, 7, 1215, 47, 4952 Investigation 2: Adding and Subtracting Rational Numbers, ACE 1822, 38(a),
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 informationSUM OF TWO SQUARES JAHNAVI BHASKAR
SUM OF TWO SQUARES JAHNAVI BHASKAR Abstract. I will investigate which numbers can be written as the sum of two squares and in how many ways, providing enough basic number theory so even the unacquainted
More informationQUADRATIC EQUATIONS EXPECTED BACKGROUND KNOWLEDGE
MODULE  1 Quadratic Equations 6 QUADRATIC EQUATIONS In this lesson, you will study aout quadratic equations. You will learn to identify quadratic equations from a collection of given equations and write
More informationMath Content
20122013 Math Content PATHWAY TO ALGEBRA I Unit Lesson Section Number and Operations in Base Ten Place Value with Whole Numbers Place Value and Rounding Addition and Subtraction Concepts Regrouping Concepts
More informationCore 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...
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 informationSummer Mathematics Packet Say Hello to Algebra 2. For Students Entering Algebra 2
Summer Math Packet Student Name: Say Hello to Algebra 2 For Students Entering Algebra 2 This summer math booklet was developed to provide students in middle school an opportunity to review grade level
More information3(vi) B. Answer: False. 3(vii) B. Answer: True
Mathematics 0N1 Solutions 1 1. Write the following sets in list form. 1(i) The set of letters in the word banana. {a, b, n}. 1(ii) {x : x 2 + 3x 10 = 0}. 3(iv) C A. True 3(v) B = {e, e, f, c}. True 3(vi)
More information1.3 Algebraic Expressions
1.3 Algebraic Expressions A polynomial is an expression of the form: a n x n + a n 1 x n 1 +... + a 2 x 2 + a 1 x + a 0 The numbers a 1, a 2,..., a n are called coefficients. Each of the separate parts,
More informationPower Series Lecture Notes
Power Series Lecture Notes A power series is a polynomial with infinitely many terms. Here is an example: $ 0ab œ â Like a polynomial, a power series is a function of. That is, we can substitute in different
More informationSquare Roots. Learning Objectives. PreActivity
Section 1. PreActivity Preparation Square Roots Our number system has two important sets of numbers: rational and irrational. The most common irrational numbers result from taking the square root of nonperfect
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 informationWelcome to Math 19500 Video Lessons. Stanley Ocken. Department of Mathematics The City College of New York Fall 2013
Welcome to Math 19500 Video Lessons Prof. Department of Mathematics The City College of New York Fall 2013 An important feature of the following Beamer slide presentations is that you, the reader, move
More informationConstruction of the Real Line 2 Is Every Real Number Rational? 3 Problems Algebra of the Real Numbers 7
About the Author v Preface to the Instructor xiii WileyPLUS xviii Acknowledgments xix Preface to the Student xxi 1 The Real Numbers 1 1.1 The Real Line 2 Construction of the Real Line 2 Is Every Real Number
More information