Skip to Content
OpenStax Logo
  1. Preface
  2. 1 Foundations
    1. Introduction
    2. 1.1 Use the Language of Algebra
    3. 1.2 Integers
    4. 1.3 Fractions
    5. 1.4 Decimals
    6. 1.5 Properties of Real Numbers
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 Solving Linear Equations
    1. Introduction
    2. 2.1 Use a General Strategy to Solve Linear Equations
    3. 2.2 Use a Problem Solving Strategy
    4. 2.3 Solve a Formula for a Specific Variable
    5. 2.4 Solve Mixture and Uniform Motion Applications
    6. 2.5 Solve Linear Inequalities
    7. 2.6 Solve Compound Inequalities
    8. 2.7 Solve Absolute Value Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Graphs and Functions
    1. Introduction
    2. 3.1 Graph Linear Equations in Two Variables
    3. 3.2 Slope of a Line
    4. 3.3 Find the Equation of a Line
    5. 3.4 Graph Linear Inequalities in Two Variables
    6. 3.5 Relations and Functions
    7. 3.6 Graphs of Functions
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Systems of Linear Equations
    1. Introduction
    2. 4.1 Solve Systems of Linear Equations with Two Variables
    3. 4.2 Solve Applications with Systems of Equations
    4. 4.3 Solve Mixture Applications with Systems of Equations
    5. 4.4 Solve Systems of Equations with Three Variables
    6. 4.5 Solve Systems of Equations Using Matrices
    7. 4.6 Solve Systems of Equations Using Determinants
    8. 4.7 Graphing Systems of Linear Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Polynomials and Polynomial Functions
    1. Introduction
    2. 5.1 Add and Subtract Polynomials
    3. 5.2 Properties of Exponents and Scientific Notation
    4. 5.3 Multiply Polynomials
    5. 5.4 Dividing Polynomials
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Factoring
    1. Introduction to Factoring
    2. 6.1 Greatest Common Factor and Factor by Grouping
    3. 6.2 Factor Trinomials
    4. 6.3 Factor Special Products
    5. 6.4 General Strategy for Factoring Polynomials
    6. 6.5 Polynomial Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 Rational Expressions and Functions
    1. Introduction
    2. 7.1 Multiply and Divide Rational Expressions
    3. 7.2 Add and Subtract Rational Expressions
    4. 7.3 Simplify Complex Rational Expressions
    5. 7.4 Solve Rational Equations
    6. 7.5 Solve Applications with Rational Equations
    7. 7.6 Solve Rational Inequalities
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Roots and Radicals
    1. Introduction
    2. 8.1 Simplify Expressions with Roots
    3. 8.2 Simplify Radical Expressions
    4. 8.3 Simplify Rational Exponents
    5. 8.4 Add, Subtract, and Multiply Radical Expressions
    6. 8.5 Divide Radical Expressions
    7. 8.6 Solve Radical Equations
    8. 8.7 Use Radicals in Functions
    9. 8.8 Use the Complex Number System
    10. Key Terms
    11. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Quadratic Equations and Functions
    1. Introduction
    2. 9.1 Solve Quadratic Equations Using the Square Root Property
    3. 9.2 Solve Quadratic Equations by Completing the Square
    4. 9.3 Solve Quadratic Equations Using the Quadratic Formula
    5. 9.4 Solve Quadratic Equations in Quadratic Form
    6. 9.5 Solve Applications of Quadratic Equations
    7. 9.6 Graph Quadratic Functions Using Properties
    8. 9.7 Graph Quadratic Functions Using Transformations
    9. 9.8 Solve Quadratic Inequalities
    10. Key Terms
    11. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Exponential and Logarithmic Functions
    1. Introduction
    2. 10.1 Finding Composite and Inverse Functions
    3. 10.2 Evaluate and Graph Exponential Functions
    4. 10.3 Evaluate and Graph Logarithmic Functions
    5. 10.4 Use the Properties of Logarithms
    6. 10.5 Solve Exponential and Logarithmic Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  12. 11 Conics
    1. Introduction
    2. 11.1 Distance and Midpoint Formulas; Circles
    3. 11.2 Parabolas
    4. 11.3 Ellipses
    5. 11.4 Hyperbolas
    6. 11.5 Solve Systems of Nonlinear Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  13. 12 Sequences, Series and Binomial Theorem
    1. Introduction
    2. 12.1 Sequences
    3. 12.2 Arithmetic Sequences
    4. 12.3 Geometric Sequences and Series
    5. 12.4 Binomial Theorem
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  14. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
  15. Index

12.1 Sequences

  • Factorial Notation
    If n is a positive integer, then n!n! is
    n!=n(n1)(n2)(3)(2)(1).n!=n(n1)(n2)(3)(2)(1).

    We define 0!0! as 1, so 0!=10!=1
  • Summation Notation
    The sum of the first n terms of a sequence whose nth term anan is written in summation notation as:
    i=1nai=a1+a2+a3+a4+a5++ani=1nai=a1+a2+a3+a4+a5++an

    The i is the index of summation and the 1 tells us where to start and the n tells us where to end.

12.2 Arithmetic Sequences

  • General Term (nth term) of an Arithmetic Sequence
    The general term of an arithmetic sequence with first term a1a1 and the common difference d is
    an=a1+(n1)dan=a1+(n1)d
  • Sum of the First n Terms of an Arithmetic Sequence
    The sum, Sn,Sn, of the first n terms of an arithmetic sequence, where a1a1 is the first term and anan is the nth term is
    Sn=n2(a1+an)Sn=n2(a1+an)

12.3 Geometric Sequences and Series

  • General Term (nth term) of a Geometric Sequence: The general term of a geometric sequence with first term a1a1 and the common ratio r is
    an=a1rn1an=a1rn1
  • Sum of the First n Terms of a Geometric Series: The sum, Sn,Sn, of the n terms of a geometric sequence is
    Sn=a1(1rn)1rSn=a1(1rn)1r

    where a1a1 is the first term and r is the common ratio.
  • Infinite Geometric Series: An infinite geometric series is an infinite sum whose first term is a1a1 and common ratio is r and is written
    a1+a1r+a1r2++a1rn1+a1+a1r+a1r2++a1rn1+
  • Sum of an Infinite Geometric Series: For an infinite geometric series whose first term is a1a1 and common ratio r,
    If|r|<1,the sum is S=a11r We say the series converges. If|r|1,the infinite geometric series does not have a sum. We say the series diverges.If|r|<1,the sum is S=a11r We say the series converges. If|r|1,the infinite geometric series does not have a sum. We say the series diverges.
  • Value of an Annuity with Interest Compounded nn Times a Year: For a principal, P, invested at the end of a compounding period, with an interest rate, r, which is compounded n times a year, the new balance, A, after t years, is
    At=P((1+rn)nt1)rnAt=P((1+rn)nt1)rn

12.4 Binomial Theorem

  • Patterns in the expansion of (a+b)n(a+b)n
    • The number of terms is n+1.n+1.
    • The first term is anan and the last term is bn.bn.
    • The exponents on a decrease by one on each term going left to right.
    • The exponents on b increase by one on each term going left to right.
    • The sum of the exponents on any term is n.
  • Pascal’s Triangle This figure shows Pascal’s Triangle. The first level is 1. The second level is 1, 1. The third level is 1, 2, 1. The fourth level is 1, 3, 3, 1. The fifth level is 1, 4, 6, 4, 1. The sixth level is 1, 5, 10, 10, 5, 1. The seventh level is 1, 6, 15, 20, 15, 6, 1
  • Binomial Coefficient (nr)(nr) : A binomial coefficient (nr),(nr), where r and n are integers with 0rn,0rn, is defined as
    (nr)=n!r!(nr)!(nr)=n!r!(nr)!

    We read (nr)(nr) as “n choose r” or “n taken r at a time”.
  • Properties of Binomial Coefficients
    (n1)=n(nn)=1(n0)=1 (n1)=n(nn)=1(n0)=1
  • Binomial Theorem: For any real numbers a, b, and positive integer n,
    (a+b)n=(n0)an+(n1)an1b1+(n2)an2b2+...+(nr)anrbr+...+(nn)bn(a+b)n=(n0)an+(n1)an1b1+(n2)an2b2+...+(nr)anrbr+...+(nn)bn
Citation/Attribution

Want to cite, share, or modify this book? This book is Creative Commons Attribution License 4.0 and you must attribute OpenStax.

Attribution information
  • If you are redistributing all or part of this book in a print format, then you must include on every physical page the following attribution:
    Access for free at https://openstax.org/books/intermediate-algebra-2e/pages/1-introduction
  • If you are redistributing all or part of this book in a digital format, then you must include on every digital page view the following attribution:
    Access for free at https://openstax.org/books/intermediate-algebra-2e/pages/1-introduction
Citation information

© Sep 2, 2020 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License 4.0 license. The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.