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DP IB Maths: AI SL

Revision Notes

Home / IB / Maths: AI SL / DP / Revision Notes / 1. Number & Algebra / 1.2 Sequences & Series / 1.2.1 Language of Sequences & Series


1.2.1 Language of Sequences & Series


Language of Sequences & Series

What is a sequence?

  • A sequence is an ordered set of numbers with a rule for finding all of the numbers in the sequence
    • For example 1, 3, 5, 7, 9, … is a sequence with the rule ‘start at one and add two to each number’
  • The numbers in a sequence are often called terms
  • The terms of a sequence are often referred to by letters with a subscript
    • In IB this will be the letter u
    • So in the sequence above, u1 = 1, u2 = 3, u3 = 5 and so on
  • Each term in a sequence can be found by substituting the term number into formula for the nth term

 

What is a series?

  • You get a series by summing up the terms in a sequence
    • E.g. For the sequence 1, 3, 5, 7, … the associated series is 1 + 3 + 5 + 7 +  …
  • We use the notation Sn to refer to the sum of the first n terms in the series
    • Sn = u1 + u2 + u3 + … + un
    • So for the series above S5 = 1 + 3 + 5 + 7 + 9 = 25

What are increasing, decreasing and periodic sequences?

  • A sequence is increasing if un+1 > un for all positive integers
    • i.e. every term is greater than the term before it
  • A sequence is decreasing if un+1 < un for all positive integers
    • i.e. every term is less than the term before it
  • A sequence is periodic if the terms repeat in a cycle

Worked Example

Determine the first five terms and the value of S5 in the sequence with terms defined by un  = 5 - 2n.

ai-sl-1-2-1-language-we-so

Sigma Notation

What is sigma notation?

  • Sigma notation is used to show the sum of a certain number of terms in a sequence
  • The symbol Σ is the capital Greek letter sigma
  • Σ stands for ‘sum’
    • The expression to the right of the Σ tells you what is being summed, and the limits above and below tell you which terms you are summing

4-5-2-sigma-not-illustr-1

  • Be careful, the limits don’t have to start with 1
    • For example sum from k space equals space 0 to 4 of left parenthesis 2 k plus 1 right parenthesis  or  sum from k space equals space 7 to 14 of left parenthesis 2 k minus 13 right parenthesis
    • r and k are commonly used variables within sigma notation

Exam Tip

  • Your GDC will be able to use sigma notation, familiarise yourself with it and practice using it to check your work

Worked Example

A sequence can be defined by  u subscript n equals space 2 space cross times space 3 to the power of n minus 1 end exponent for  n element of space straight integer numbers to the power of plus .

 

a)
Write an expression for u subscript 1 space plus space u subscript 2 space plus space u subscript 3 space plus space... space plus space u subscript 6 using sigma notation.

ai-sl-1-2-1-sigma-a

 

b)
Write an expression for u subscript 7 space plus space u subscript 8 space plus space u subscript 9 space plus space... space plus space u subscript 12using sigma notation.

ai-sl-1-2-1-sigma-b



  • 1. Number & Algebra
    • 1.1 Number Toolkit
      • 1.1.1 Standard Form
        • 1.1.2 Exponents & Logarithms
          • 1.1.3 Approximation & Estimation
            • 1.1.4 GDC: Solving Equations
            • 1.2 Sequences & Series
              • 1.2.1 Language of Sequences & Series
                • 1.2.2 Arithmetic Sequences & Series
                  • 1.2.3 Geometric Sequences & Series
                    • 1.2.4 Applications of Sequences & Series
                    • 1.3 Financial Applications
                      • 1.3.1 Compound Interest & Depreciation
                        • 1.3.2 Amortisation & Annuities
                      • 2. Functions
                        • 2.1 Linear Functions & Graphs
                          • 2.1.1 Equations of a Straight Line
                          • 2.2 Further Functions & Graphs
                            • 2.2.1 Functions
                              • 2.2.2 Graphing Functions
                                • 2.2.3 Properties of Graphs
                                • 2.3 Modelling with Functions
                                  • 2.3.1 Linear & Piecewise Models
                                    • 2.3.2 Quadratic & Cubic Models
                                      • 2.3.3 Exponential Models
                                        • 2.3.4 Direct & Inverse Variation
                                          • 2.3.5 Sinusoidal Models
                                            • 2.3.6 Strategy for Modelling Functions
                                          • 3. Geometry & Trigonometry
                                            • 3.1 Geometry Toolkit
                                              • 3.1.1 Coordinate Geometry
                                                • 3.1.2 Arcs & Sectors
                                                • 3.2 Geometry of 3D Shapes
                                                  • 3.2.1 3D Coordinate Geometry
                                                    • 3.2.2 Volume & Surface Area
                                                    • 3.3 Trigonometry
                                                      • 3.3.1 Pythagoras & Right-Angled Triganometry
                                                        • 3.3.2 Non Right-Angled Trigonometry
                                                          • 3.3.3 Applications of Trigonometry & Pythagoras
                                                          • 3.4 Voronoi Diagrams
                                                            • 3.4.1 Voronoi Diagrams
                                                              • 3.4.2 Toxic Waste Dump Problem
                                                            • 4. Statistics & Probability
                                                              • 4.1 Statistics Toolkit
                                                                • 4.1.1 Sampling & Data Collection
                                                                  • 4.1.2 Statistical Measures
                                                                    • 4.1.3 Frequency Tables
                                                                      • 4.1.4 Linear Transformations of Data
                                                                        • 4.1.5 Outliers
                                                                          • 4.1.6 Univariate Data
                                                                            • 4.1.7 Interpreting Data
                                                                            • 4.2 Correlation & Regression
                                                                              • 4.2.1 Bivariate data
                                                                                • 4.2.2 Correlation Coefficients
                                                                                  • 4.2.3 Linear Regression
                                                                                  • 4.3 Probability
                                                                                    • 4.3.1 Probability & Types of Events
                                                                                      • 4.3.2 Conditional Probability
                                                                                        • 4.3.3 Sample Space Diagrams
                                                                                        • 4.4 Probability Distributions
                                                                                          • 4.4.1 Discrete Probability Distributions
                                                                                            • 4.4.2 Expected Values
                                                                                            • 4.5 Binomial Distribution
                                                                                              • 4.5.1 The Binomial Distribution
                                                                                                • 4.5.2 Calculating Binomial Probabilities
                                                                                                • 4.6 Normal Distribution
                                                                                                  • 4.6.1 The Normal Distribution
                                                                                                    • 4.6.2 Calculations with Normal Distribution
                                                                                                    • 4.7 Hypothesis Testing
                                                                                                      • 4.7.1 Hypothesis Testing
                                                                                                        • 4.7.2 Chi-squared Test for Independence
                                                                                                          • 4.7.3 Goodness of Fit Test
                                                                                                            • 4.7.4 The t-test
                                                                                                          • 5. Calculus
                                                                                                            • 5.1 Differentiation
                                                                                                              • 5.1.1 Introduction to Differentiation
                                                                                                                • 5.1.2 Applications of Differentiation
                                                                                                                  • 5.1.3 Modelling with Differentiation
                                                                                                                  • 5.2 Integration
                                                                                                                    • 5.2.1 Trapezoid Rule: Numerical Integration
                                                                                                                      • 5.2.2 Introduction to Integration
                                                                                                                        • 5.2.3 Applications of Integration
                                                                                                                      Amber Perry

                                                                                                                      Author: Amber

                                                                                                                      Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.


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