Exam #5 Sequences and Series


 Ross Elliott
 2 years ago
 Views:
Transcription
1 College of the Redwoods Mathematics Department Math 30 College Algebra Exam #5 Sequences and Series David Arnold Don Hickethier Copyright c 000 Last Revision Date: April 9, 004 Version.00
2 Multiple Choice Questions Directions: In each of the following exercises, select the best answer and darken the corresponding oval on your scantron sheet.. Find the n th term of the sequence, 4, 9, 6,.... (a) a n = 3 (b) a n = n (c) a n = ( ) n (d) a n = ( ) n (n) (e) a n = n. Find the 5 th term of the sequence a n = (n ) 3. (a) 3 (b) 47 (c) 9 (d) 5 (e) Find the 5 th term of the recursive sequence where a =. a k+ = a k (a) 7 (b) 0 (c) 9 (d) 5 (e) 4. Find the 8 th term of the recursive sequence where a = and a = 3. a k+ = a k + a k (a) 34 (b) 55 (c) 99 (d) 0 (e) 9 5. Compute the sum 3 k= k. (a) 3 (b) 6 (c) 3 5 (d) (e) 6 6. Find the n th term of the arithmetic sequence 8, 5,,, 4,.... (a) a n = 8 + 3(n ) (b) a n = 8(3) n (c) a n = 8 3n (d) a n = 8 n (e) a n = 3 8(n ) 7. Find the twohundredth term, a 00, of the sequence, 5, 8,,.... (a) 399 (b) 499 (c) 599 (d) 603 (e) 583
3 8. The first term of an arithmetic sequence is a = 3 and the eleventh term is a = 3. Find the n th term, a n, of the sequence. (a) a n = 3 + 3(n ) (b) a n = 3 + 0(n ) (c) a n = 3 + (n ) (d) a n = 3 + 0(n ) (e) a n = 3 + (n ) 9. Calculate the sum 00 k + 5. k= (a) 8,500 (b),700 (c) 9,00 (d) 38,500 (e) 4,00 0. Calculate the sum of the arithmetic series (a) 4 (b) 953 (c) 3906 (d) 7560 (e) Find the common ratio of the geometric sequence 4,,,... (a) (b) 4 (c) 4 (d) (e). The n th term of the geometric sequence 3, 6,, 4,... is (a) a n = 3( ) n (b) a n = ( 3) n (c) a n = (3) n (d) a n = 3( ) n (e) a n = 3() n 3. Find the sum of the first 8 terms of the geometric sequence,, 4, 8,... (a) 55 (b) 6 (c) 56 (d) 8 (e) 8 4. Compute the sum + a + a + a 3 + a 4 + a 5. (a) a6 (b) a6 a a (d) a5 (e) a a 5. Find the sum of the infinite series ( ) k 5. 3 (c) + a5 + a (a) 0 (b) 0 3 (c) 5 (d) 5 (e)
4 6. Find the sum of the infinite series + x + x + x 3 + given < x <. (a) xn x (b) (c) (d) x x x 7. Compute n C 3. (a) n! (b) n 6 3 n(n )(n ) (d) (e) n(n 3) 6 8. Solve for n. ( ) ( ) n + n = x (e) (c) n(n )(n ) (a) 9 (b) 8 (c) 7 (d) 0 (e) 6 9. Calculate the next row of Pascal s Triangle. 3 3 (a) (b) (c) (d) (e) Use the binomial theorem to expand (x y) 4. x (a) 6x 4 y 4 (b) 6x 4 3x 3 y + 4x y 8xy 3 + y 4 (c) 6x 4 8xy + y 4 (d) x 4 4x 3 y + 6x y 4xy 3 + y 4 (e) 6x 4 8x 3 y + x y 8xy 3 + y 4. Find the th term of (3x + y). (a) y (b) 44x y 9 (c) 56xy 0 (d) 33xy 0 (e) 4x y 9. One of the solutions of is 4 ( ) 4 (x) 4 k ( ) k = k (a) 0 (b) (c) (d) 4 (e) 3. Simplify (a) 5 (b) 7 (c) 9 (d) 30 (e) 3
5 Solutions to Multiple Choice Questions Solution to Question : The all of the terms of the sequence are perfect squares and the terms alternate sign. So the n th term of the sequence is a n = ( ) n (n). Solution to Question : To find the 5 th term of the sequence replace n with 5 in the sequence a n = (n ) 3. a 5 = (5 ) 3 = (4) 3 = 3 3 = 9 Solution to Question 3: The sequence is defined recursively by a k+ = a k. a = a = a = () = 3 a 3 = a = (3) = 5 a 4 = a 3 = (5) = 9 a 5 = a 4 = (9) = 7 Solution to Question 4: The sequence is defined recursively by a k+ = a k + a k with the first two terms given. Thus Solution to Question 5: 3 k= a = a = 3 a 3 = a + a = + 3 = 5 a 4 = a 3 + a = = 8 a 5 = a 4 + a 3 = = 3 a 6 = a 5 + a 4 = = a 7 = a 6 + a 5 = 3 + = 34 a 8 = a 7 + a 6 = + 34 = 55 k = = = 6 6 Solution to Question 6: The first term of the arithmetic series is a = 8 and the common difference is d = 3. Using the fact that for an arithmetic series the n th term is a n = a + (n )d, a n = 8 + 3(n )
6 Solution to Question 7: The sequence is arithmetic with first term a = 3 and common difference d = 3. Then a n = + (n )3 a n = + 3n 3 a n = 3n a 00 = 3(00) a 00 = 599 Solution to Question 8: Use the fact that for an arithmetic series the n th term is a n = a +(n )d to solve for d with a = 3 and a = 3. 3 = 3 + ( )d 0 = 0d = d Therefore, a n = 3 + (n ). Solution to Question 9: Write the series to notice the sum is an arithmetic series. 00 k + 5 = k= Indeed the sum is arithmetic with first term a = 7 and last term a 00 = 405. Now use the arithmetic series formula S n = n a + a n. 00 k + 5 = 00 ( ) = 00(4) = 4, 000 k= Solution to Question 0: The problem gives the first and last terms of the arithmetic series. To use the arithmetic series formula, S n = n (a + a n ), the number of terms needs to be determined. The first term is a = 3 and the common difference is d = 4, so a n = 3 + 4(n ). Thus Now it is possible to compute the sum. 3 = 3 + 4(n ) 0 = 4(n ) 30 = n 3 = n S 3 = 3 (3 + 3) = 3 (6) = 3(63) = 953
7 Solution to Question : The common ratio is r = a k+ a k = 4 = = Solution to Question : Use the n th term formula a n = a r n with a first term a = 3 and common ratio r = 6 3 = to get a n = 3( ) n. Solution to Question 3: The finite sum for a geometric series is S n = a ( r n ) where a r is the first term and r is the common ratio. The first term is a = and the common ratio is r = = 4 =. The sum of the first 8 terms is S 8 = ( 8 ) = 56 = 55 Solution to Question 4: The finite sum for a geometric series is S n = a ( r n ) r first term is a = and the common ratio is r = a = a. The sum of the 6 terms is where the S 6 = ( a6 ) a Solution to Question 5: The infinite sum for a geometric series is S = a r where a is the first term and r is the common ratio such that r <. The first term is a = 5 and the common ratio is r =. Since the common ratio is less than the infinite sum is 3 S = 5 /3 = 5 /3 = 5 Solution to Question 6: The infinite sum for a geometric series is S = a r where a is the first term and r is the common ratio such that r <. The first term is a = and the common ratio is r = x = x. Since < x < the infinite sum is S = x
8 Solution to Question 7: The number of combinations of n objects taken k at a time is nc k = n! (n k)!k!. So, nc 3 = Solution to Question 8: n! n(n )(n )(n 3)! n(n )(n ) = = (n 3)!3! 3!(n 3)! 6 ( ) ( ) n + n = (n + )! 5n! = 4!(n + 4)! 3!(n 3)! (n + )n! 5n! = 4(n 3)! 6(n 3)! n + 4 = 5 6 n + = 0 n = 9 Solution to Question 9: The next row of Pascal s Triangle is Solution to Question 0: The binomial expansion for (a + b) 4 is (a + b) 4 = a 4 + 4a 3 b + 6a b + 4ab 3 + b 4. To evaluate (x y) 4 let a = x and b = y in the equation above. (x y) 4 = (x) 4 + 4(x) 3 ( y) + 6(x) ( y) + 4(x)( y) 3 + ( y) 4 = 6x 4 3x 3 y + 4x y 8xy 3 + y 4 Solution to Question : The (k + ) st term in the binomial expansion of (a + b) n is ( n k) a n k b k. So the th term of the binomial expansion of (3x + y) is ( ) (3x) (y) 0 = (3x)y 0 = 33xy 0 0
9 Solution to Question : To solve 4 ( 4 k) (x) 4 k ( ) k = notice the left hand side of the equation is the fourth degree binomial expansion for (x ). That is 4 ( ) 4 (x) 4 k ( ) k = k (x ) 4 = (x ) = ± x = 3 x = Solution to Question 3: The sum resembles the sigma notation for the binomial expansion, (a + b) n = n ( n k) (a) n k (b) k with n = 5. If a = and b =, then = 5 = ( + ) 5 = 3 () n k () k k
AFM Ch.12  Practice Test
AFM Ch.2  Practice Test Multiple Choice Identify the choice that best completes the statement or answers the question.. Form a sequence that has two arithmetic means between 3 and 89. a. 3, 33, 43, 89
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 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 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 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 information0018 DATA ANALYSIS, PROBABILITY and STATISTICS
008 DATA ANALYSIS, PROBABILITY and STATISTICS A permutation tells us the number of ways we can combine a set where {a, b, c} is different from {c, b, a} and without repetition. r is the size of of the
More informationAlgebra 12. A. Identify and translate variables and expressions.
St. Mary's College High School Algebra 12 The Language of Algebra What is a variable? A. Identify and translate variables and expressions. The following apply to all the skills How is a variable used
More informationCOWLEY COUNTY COMMUNITY COLLEGE REVIEW GUIDE Compass Algebra Level 2
COWLEY COUNTY COMMUNITY COLLEGE REVIEW GUIDE Compass Algebra Level This study guide is for students trying to test into College Algebra. There are three levels of math study guides. 1. If x and y 1, what
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 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 informationThinkwell s Homeschool Algebra 2 Course Lesson Plan: 34 weeks
Thinkwell s Homeschool Algebra 2 Course Lesson Plan: 34 weeks Welcome to Thinkwell s Homeschool Algebra 2! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson
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 informationALGEBRA 2/TRIGONOMETRY
ALGEBRA /TRIGONOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION ALGEBRA /TRIGONOMETRY Thursday, January 9, 015 9:15 a.m to 1:15 p.m., only Student Name: School Name: The possession
More informationMath 313 Lecture #10 2.2: The Inverse of a Matrix
Math 1 Lecture #10 2.2: The Inverse of a Matrix Matrix algebra provides tools for creating many useful formulas just like real number algebra does. For example, a real number a is invertible if there is
More informationSOL WarmUp Graphing Calculator Active
A.2a (a) Using laws of exponents to simplify monomial expressions and ratios of monomial expressions 1. Which expression is equivalent to (5x 2 )(4x 5 )? A 9x 7 B 9x 10 C 20x 7 D 20x 10 2. Which expression
More informationALGEBRA 2/TRIGONOMETRY
ALGEBRA /TRIGONOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION ALGEBRA /TRIGONOMETRY Tuesday, June 1, 011 1:15 to 4:15 p.m., only Student Name: School Name: Print your name
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 informationSection 62 Mathematical Induction
6 Mathematical Induction 457 In calculus, it can be shown that e x k0 x k k! x x x3!! 3!... xn n! where the larger n is, the better the approximation. Problems 6 and 6 refer to this series. Note that
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 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 informationALGEBRA 2/TRIGONOMETRY
ALGEBRA /TRIGONOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION ALGEBRA /TRIGONOMETRY Tuesday, January 8, 014 1:15 to 4:15 p.m., only Student Name: School Name: The possession
More informationAlgebra Unpacked Content For the new Common Core standards that will be effective in all North Carolina schools in the 201213 school year.
This document is designed to help North Carolina educators teach the Common Core (Standard Course of Study). NCDPI staff are continually updating and improving these tools to better serve teachers. Algebra
More informationPrecalculus REVERSE CORRELATION. Content Expectations for. Precalculus. Michigan CONTENT EXPECTATIONS FOR PRECALCULUS CHAPTER/LESSON TITLES
Content Expectations for Precalculus Michigan Precalculus 2011 REVERSE CORRELATION CHAPTER/LESSON TITLES Chapter 0 Preparing for Precalculus 01 Sets There are no statemandated Precalculus 02 Operations
More informationTaylor and Maclaurin Series
Taylor and Maclaurin Series In the preceding section we were able to find power series representations for a certain restricted class of functions. Here we investigate more general problems: Which functions
More informationWorksheet on induction Calculus I Fall 2006 First, let us explain the use of for summation. The notation
Worksheet on induction MA113 Calculus I Fall 2006 First, let us explain the use of for summation. The notation f(k) means to evaluate the function f(k) at k = 1, 2,..., n and add up the results. In other
More informationCofactor Expansion: Cramer s Rule
Cofactor Expansion: Cramer s Rule MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Introduction Today we will focus on developing: an efficient method for calculating
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 informationStanford Math Circle: Sunday, May 9, 2010 SquareTriangular Numbers, Pell s Equation, and Continued Fractions
Stanford Math Circle: Sunday, May 9, 00 SquareTriangular 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 information4. Binomial Expansions
4. Binomial Expansions 4.. Pascal's Triangle The expansion of (a + x) 2 is (a + x) 2 = a 2 + 2ax + x 2 Hence, (a + x) 3 = (a + x)(a + x) 2 = (a + x)(a 2 + 2ax + x 2 ) = a 3 + ( + 2)a 2 x + (2 + )ax 2 +
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 informationGrade 7/8 Math Circles Sequences and Series
Faculty of Mathematics Waterloo, Ontario N2L 3G1 Centre for Education in Mathematics and Computing Grade 7/8 Math Circles Sequences and Series November 30, 2012 What are sequences? A sequence is an ordered
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 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 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 informationFactoring (pp. 1 of 4)
Factoring (pp. 1 of 4) Algebra Review Try these items from middle school math. A) What numbers are the factors of 4? B) Write down the prime factorization of 7. C) 6 Simplify 48 using the greatest common
More informationFriday, January 29, 2016 9:15 a.m. to 12:15 p.m., only
ALGEBRA /TRIGONOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION ALGEBRA /TRIGONOMETRY Friday, January 9, 016 9:15 a.m. to 1:15 p.m., only Student Name: School Name: The possession
More informationMath 1050 Khan Academy Extra Credit Algebra Assignment
Math 1050 Khan Academy Extra Credit Algebra Assignment KhanAcademy.org offers over 2,700 instructional videos, including hundreds of videos teaching algebra concepts, and corresponding problem sets. In
More informationCollege Algebra. Barnett, Raymond A., Michael R. Ziegler, and Karl E. Byleen. College Algebra, 8th edition, McGrawHill, 2008, ISBN: 9780072867381
College Algebra Course Text Barnett, Raymond A., Michael R. Ziegler, and Karl E. Byleen. College Algebra, 8th edition, McGrawHill, 2008, ISBN: 9780072867381 Course Description This course provides
More informationk, then n = p2α 1 1 pα k
Powers of Integers An integer n is a perfect square if n = m for some integer m. Taking into account the prime factorization, if m = p α 1 1 pα k k, then n = pα 1 1 p α k k. That is, n is a perfect square
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 informationMATH 65 NOTEBOOK CERTIFICATIONS
MATH 65 NOTEBOOK CERTIFICATIONS Review Material from Math 60 2.5 4.3 4.4a Chapter #8: Systems of Linear Equations 8.1 8.2 8.3 Chapter #5: Exponents and Polynomials 5.1 5.2a 5.2b 5.3 5.4 5.5 5.6a 5.7a 1
More informationUtah Core Curriculum for Mathematics
Core Curriculum for Mathematics correlated to correlated to 2005 Chapter 1 (pp. 2 57) Variables, Expressions, and Integers Lesson 1.1 (pp. 5 9) Expressions and Variables 2.2.1 Evaluate algebraic expressions
More informationProbability and Statistics Vocabulary List (Definitions for Middle School Teachers)
Probability and Statistics Vocabulary List (Definitions for Middle School Teachers) B Bar graph a diagram representing the frequency distribution for nominal or discrete data. It consists of a sequence
More informationThe thing that started it 8.6 THE BINOMIAL THEOREM
476 Chapter 8 Discrete Mathematics: Functions on the Set of Natural Numbers (b) Based on your results for (a), guess the minimum number of moves required if you start with an arbitrary number of n disks.
More informationLAKE ELSINORE UNIFIED SCHOOL DISTRICT
LAKE ELSINORE UNIFIED SCHOOL DISTRICT Title: PLATO Algebra 1Semester 2 Grade Level: 1012 Department: Mathematics Credit: 5 Prerequisite: Letter grade of F and/or N/C in Algebra 1, Semester 2 Course Description:
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 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 informationMATH. ALGEBRA I HONORS 9 th Grade 12003200 ALGEBRA I HONORS
* Students who scored a Level 3 or above on the Florida Assessment Test Math Florida Standards (FSAMAFS) are strongly encouraged to make Advanced Placement and/or dual enrollment courses their first choices
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 noncourse based remediation in developmental mathematics. This structure will
More informationSome sequences have a fixed length and have a last term, while others go on forever.
Sequences and series Sequences A sequence is a list of numbers (actually, they don t have to be numbers). Here is a sequence: 1, 4, 9, 16 The order makes a difference, so 16, 9, 4, 1 is a different sequence.
More informationm i: is the mass of each particle
Center of Mass (CM): The center of mass is a point which locates the resultant mass of a system of particles or body. It can be within the object (like a human standing straight) or outside the object
More informationDifference of Squares and Perfect Square Trinomials
4.4 Difference of Squares and Perfect Square Trinomials 4.4 OBJECTIVES 1. Factor a binomial that is the difference of two squares 2. Factor a perfect square trinomial In Section 3.5, we introduced some
More informationInstructions for SA Completion
Instructions for SA Completion 1 Take notes on these Pythagorean Theorem Course Materials then do and check the associated practice questions for an explanation on how to do the Pythagorean Theorem Substantive
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 informationx if x 0, x if x < 0.
Chapter 3 Sequences In this chapter, we discuss sequences. We say what it means for a sequence to converge, and define the limit of a convergent sequence. We begin with some preliminary results about the
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 informationAlgebra I Credit Recovery
Algebra I Credit Recovery COURSE DESCRIPTION: The purpose of this course is to allow the student to gain mastery in working with and evaluating mathematical expressions, equations, graphs, and other topics,
More informationMultiplying Polynomials 5
Name: Date: Start Time : End Time : Multiplying Polynomials 5 (WS#A10436) Polynomials are expressions that consist of two or more monomials. Polynomials can be multiplied together using the distributive
More informationCM2202: Scientific Computing and Multimedia Applications General Maths: 2. Algebra  Factorisation
CM2202: Scientific Computing and Multimedia Applications General Maths: 2. Algebra  Factorisation Prof. David Marshall School of Computer Science & Informatics Factorisation Factorisation is a way of
More informationLECTURE 2. Part/ References Topic/Sections Notes/Speaker. 1 Enumeration 1 1.1 Generating functions... 1. Combinatorial. Asst #1 Due FS A.
Wee Date Sections f aculty of science MATH 8954 Fall 0 Contents 1 Sept 7 I1, I, I3 Symbolic methods I A3/ C Introduction to Prob 4 A sampling 18 IX1 of counting sequences Limit Laws and and generating
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 informationMATH 2 Course Syllabus Spring Semester 2007 Instructor: Brian Rodas
MATH 2 Course Syllabus Spring Semester 2007 Instructor: Brian Rodas Class Room and Time: MC83 MTWTh 2:15pm3:20pm Office Room: MC38 Office Phone: (310)4348673 Email: rodas brian@smc.edu Office Hours:
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 informationAdministrative  Master Syllabus COVER SHEET
Administrative  Master Syllabus COVER SHEET Purpose: It is the intention of this to provide a general description of the course, outline the required elements of the course and to lay the foundation for
More informationFactoring Special Polynomials
6.6 Factoring Special Polynomials 6.6 OBJECTIVES 1. Factor the difference of two squares 2. Factor the sum or difference of two cubes In this section, we will look at several special polynomials. These
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 informationNSM100 Introduction to Algebra Chapter 5 Notes Factoring
Section 5.1 Greatest Common Factor (GCF) and Factoring by Grouping Greatest Common Factor for a polynomial is the largest monomial that divides (is a factor of) each term of the polynomial. GCF is the
More informationAlum Rock Elementary Union School District Algebra I Study Guide for Benchmark III
Alum Rock Elementary Union School District Algebra I Study Guide for Benchmark III Name Date Adding and Subtracting Polynomials Algebra Standard 10.0 A polynomial is a sum of one ore more monomials. Polynomial
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 informationMath 141. Lecture 7: Variance, Covariance, and Sums. Albyn Jones 1. 1 Library 304. jones/courses/141
Math 141 Lecture 7: Variance, Covariance, and Sums Albyn Jones 1 1 Library 304 jones@reed.edu www.people.reed.edu/ jones/courses/141 Last Time Variance: expected squared deviation from the mean: Standard
More informationALGEBRA 2/TRIGONOMETRY
ALGEBRA /TRIGONOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION ALGEBRA /TRIGONOMETRY Friday, June 14, 013 1:15 to 4:15 p.m., only Student Name: School Name: The possession
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 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 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 informationSYSTEMS OF PYTHAGOREAN TRIPLES. Acknowledgements. I would like to thank Professor Laura Schueller for advising and guiding me
SYSTEMS OF PYTHAGOREAN TRIPLES CHRISTOPHER TOBINCAMPBELL Abstract. This paper explores systems of Pythagorean triples. It describes the generating formulas for primitive Pythagorean triples, determines
More informationBookTOC.txt. 1. Functions, Graphs, and Models. Algebra Toolbox. Sets. The Real Numbers. Inequalities and Intervals on the Real Number Line
College Algebra in Context with Applications for the Managerial, Life, and Social Sciences, 3rd Edition Ronald J. Harshbarger, University of South Carolina  Beaufort Lisa S. Yocco, Georgia Southern University
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 informationArithmetic Operations. The real numbers have the following properties: In particular, putting a 1 in the Distributive Law, we get
Review of Algebra REVIEW OF ALGEBRA Review of Algebra Here we review the basic rules and procedures of algebra that you need to know in order to be successful in calculus. Arithmetic Operations The real
More informationSECTION 105 Multiplication Principle, Permutations, and Combinations
105 Multiplication Principle, Permutations, and Combinations 761 54. Can you guess what the next two rows in Pascal s triangle, shown at right, are? Compare the numbers in the triangle with the binomial
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 informationSince the ratios are constant, the sequence is geometric. The common ratio is.
Determine whether each sequence is arithmetic, geometric, or neither. Explain. 1. 200, 40, 8, Since the ratios are constant, the sequence is geometric. The common ratio is. 2. 2, 4, 16, The ratios are
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 informationAlgebra 1 Course Title
Algebra 1 Course Title Course wide 1. What patterns and methods are being used? Course wide 1. Students will be adept at solving and graphing linear and quadratic equations 2. Students will be adept
More informationModuMath Algebra Lessons
ModuMath Algebra Lessons Program Title 1 Getting Acquainted With Algebra 2 Order of Operations 3 Adding & Subtracting Algebraic Expressions 4 Multiplying Polynomials 5 Laws of Algebra 6 Solving Equations
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 informationMATH 095, College Prep Mathematics: Unit Coverage Prealgebra topics (arithmetic skills) offered through BSE (Basic Skills Education)
MATH 095, College Prep Mathematics: Unit Coverage Prealgebra topics (arithmetic skills) offered through BSE (Basic Skills Education) Accurately add, subtract, multiply, and divide whole numbers, integers,
More informationALGEBRA 2 CRA 2 REVIEW  Chapters 16 Answer Section
ALGEBRA 2 CRA 2 REVIEW  Chapters 16 Answer Section MULTIPLE CHOICE 1. ANS: C 2. ANS: A 3. ANS: A OBJ: 53.1 Using Vertex Form SHORT ANSWER 4. ANS: (x + 6)(x 2 6x + 36) OBJ: 64.2 Solving Equations by
More informationHigher Education Math Placement
Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication
More informationAlgebra Unit 6 Syllabus revised 2/27/13 Exponents and Polynomials
Algebra Unit 6 Syllabus revised /7/13 1 Objective: Multiply monomials. Simplify expressions involving powers of monomials. Preassessment: Exponents, Fractions, and Polynomial Expressions Lesson: Pages
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 information( ) FACTORING. x In this polynomial the only variable in common to all is x.
FACTORING Factoring is similar to breaking up a number into its multiples. For example, 10=5*. The multiples are 5 and. In a polynomial it is the same way, however, the procedure is somewhat more complicated
More informationSouth Carolina College and CareerReady (SCCCR) PreCalculus
South Carolina College and CareerReady (SCCCR) PreCalculus Key Concepts Arithmetic with Polynomials and Rational Expressions PC.AAPR.2 PC.AAPR.3 PC.AAPR.4 PC.AAPR.5 PC.AAPR.6 PC.AAPR.7 Standards Know
More informationwith functions, expressions and equations which follow in units 3 and 4.
Grade 8 Overview View unit yearlong overview here The unit design was created in line with the areas of focus for grade 8 Mathematics as identified by the Common Core State Standards and the PARCC Model
More informationDELAWARE MATHEMATICS CONTENT STANDARDS GRADES 910. PAGE(S) WHERE TAUGHT (If submission is not a book, cite appropriate location(s))
Prentice Hall University of Chicago School Mathematics Project: Advanced Algebra 2002 Delaware Mathematics Content Standards (Grades 910) STANDARD #1 Students will develop their ability to SOLVE PROBLEMS
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 informationA Concrete Introduction. to the Abstract Concepts. of Integers and Algebra using Algebra Tiles
A Concrete Introduction to the Abstract Concepts of Integers and Algebra using Algebra Tiles Table of Contents Introduction... 1 page Integers 1: Introduction to Integers... 3 2: Working with Algebra Tiles...
More informationCPM Educational Program
CPM Educational Program A California, NonProfit Corporation Chris Mikles, National Director (888) 8084276 email: mikles @cpm.org CPM Courses and Their Core Threads Each course is built around a few
More informationThe program also provides supplemental modules on topics in geometry and probability and statistics.
Algebra 1 Course Overview Students develop algebraic fluency by learning the skills needed to solve equations and perform important manipulations with numbers, variables, equations, and inequalities. Students
More informationAdvanced Math Study Guide
Advanced Math Study Guide Topic Finding Triangle Area (Ls. 96) using A=½ bc sin A (uses Law of Sines, Law of Cosines) Law of Cosines, Law of Cosines (Ls. 81, Ls. 72) Finding Area & Perimeters of Regular
More information