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DP IB Maths: AA SL

Revision Notes

Home / IB / Maths: AA SL / DP / Revision Notes / 1. Number & Algebra / 1.3 Sequences & Series / 1.3.3 Geometric Sequences & Series


1.3.3 Geometric Sequences & Series


Geometric Sequences

What is a geometric sequence?

  • In a geometric sequence, there is a common ratio, r, between consecutive terms in the sequence
    • For example, 2, 6, 18, 54, 162, … is a sequence with the rule ‘start at two and multiply each number by three’
      • The first term, u1, is 2
      • The common ratio, r, is 3
  • A geometric sequence can be increasing (r > 1) or decreasing (0 < r < 1)
  •  If the common ratio is a negative number the terms will alternate between positive and negative values
    • For example, 1, -4, 16, -64, 256, … is a sequence with the rule ‘start at one and multiply each number by negative four’
        • The first term, u1, is 1
        • The common ratio, r, is -4
  • Each term of a geometric sequence is referred to by the letter u with a subscript determining its place in the sequence

How do I find a term in a geometric sequence?

  • The n to the power of t h end exponent term formula for a geometric sequence is given as

u subscript n equals u subscript 1 r to the power of n minus 1 end exponent

    • Where u subscript 1 is the first term, and r is the common ratio
    • This formula allows you to find any term in the geometric sequence
    • It is given in the formula booklet, you do not need to know how to derive it
  • Enter the information you have into the formula and use your GDC to find the value of the term
  • Sometimes you will be given a term and asked to find the first term or the common ratio
    • Substitute the information into the formula and solve the equation
      • You could use your GDC for this
  • Sometimes you will be given two or more consecutive terms and asked to find both the first term and the common ratio
    • Find the common ratio by dividing a term by the one before it
    • Substitute this and one of the terms into the formula to find the first term
  • Sometimes you may be given a term and the formula for the nth term and asked to find the value of n
    • You can solve these using logarithms on your GDC

 

Exam Tip

  • You will sometimes need to use logarithms to answer geometric sequences questions 
    • Make sure you are confident doing this
    • Practice using your GDC for different types of questions

Worked Example

The sixth term, u subscript 6, of a geometric sequence is 486 and the seventh term, u subscript 7, is 1458. 

Find,

i)
the common ratio, r, of the sequence,

ai-sl-1-2-3-geo-seq-i

 

ii)
the first term of the sequence, u subscript 1.

ai-sl-1-2-3-geo-seq-ii

Geometric Series

How do I find the sum of a geometric series?

  • A geometric series is the sum of a certain number of terms in a geometric sequence
    • For the geometric sequence 2, 6, 18, 54, … the geometric series is 2 + 6 + 18 + 54 + …
  • The following formulae will let you find the sum of the first n terms of a geometric series:

S subscript n equals fraction numerator u subscript 1 left parenthesis r to the power of n minus 1 right parenthesis over denominator r minus 1 end fraction equals space fraction numerator u subscript 1 left parenthesis 1 minus r to the power of n right parenthesis over denominator 1 minus r end fraction

      • u subscript 1 is the first term
      • r is the common ratio
    • Both formulae are given in the formula booklet, you do not need to know how to derive them
  • You can use whichever formula is more convenient for a given question
    • The first version of the formula is more convenient if r space greater than space 1 and the second is more convenient if r space less than space 1
  • A question will often give you the sum of a certain number of terms and ask you to find the value of the first term, the common ratio, or the number of terms within the sequence
    • Substitute the information into the formula and solve the equation
      • You could use your GDC for this

Exam Tip

  • The geometric series formulae are in the formulae booklet, you don't need to memorise them
    • Make sure you can locate them quickly in the formula booklet

Worked Example

A geometric sequence has u subscript 1 space equals space 25 and r space equals space 0.8.  Find the value of u subscript 5 and S subscript 5.

ai-sl-1-2-3-geo-series

Sum to Infinity

What is the sum to infinity of a geometric series?

  • A geometric sequence will either increase or decrease away from zero or the terms will get progressively closer to zero
    • Terms will get closer to zero if the common ratio, r, is between 1 and -1
  • If the terms are getting closer to zero then the series is said to converge
    • This means that the sum of the series will approach a limiting value
    • As the number of terms increase, the sum of the terms will get closer to the limiting value

 

How do we calculate the sum to infinity?

  • If asked to find out if a geometric sequence converges find the value of r
    • If vertical line r vertical line space space less than space 1 space  then the sequence converges
    • If vertical line r vertical line space space greater or equal than space 1 space then the sequence does not converge and the sum to infinity cannot be calculated
    • vertical line r vertical line space less than space 1 spacemeans negative 1 space less than space r space less than space 1 space
  • If vertical line r vertical line space less than space 1, then the geometric series converges to a finite value given by the formula

S subscript infinity equals fraction numerator u subscript 1 over denominator 1 minus r end fraction space comma space blank open vertical bar r close vertical bar less than 1

    • u subscript 1 is the first term
    • r is the common ratio
    • This is in the formula book, you do not need to remember it

Exam Tip

  • Learn and remember the conditions for when a sum to infinity can be calculated

Worked Example

The first three terms of a geometric sequence are  6 space comma space space 2 space comma space space 2 over 3.  Explain why the series converges and find the sum to infinity.

1-3-3-aa-sl-sum-to-infinity-we-solution-



  • 1. Number & Algebra
    • 1.1 Number Toolkit
      • 1.1.1 Standard Form
        • 1.1.2 Laws of Indices
        • 1.2 Exponentials & Logs
          • 1.2.1 Introduction to Logarithms
            • 1.2.2 Laws of Logarithms
              • 1.2.3 Solving Exponential Equations
              • 1.3 Sequences & Series
                • 1.3.1 Language of Sequences & Series
                  • 1.3.2 Arithmetic Sequences & Series
                    • 1.3.3 Geometric Sequences & Series
                      • 1.3.4 Applications of Sequences & Series
                        • 1.3.5 Compound Interest & Depreciation
                        • 1.4 Proof & Reasoning
                          • 1.4.1 Proof
                          • 1.5 Binomial Theorem
                            • 1.5.1 Binomial Theorem
                          • 2. Functions
                            • 2.1 Linear Functions & Graphs
                              • 2.1.1 Equations of a Straight Line
                              • 2.2 Quadratic Functions & Graphs
                                • 2.2.1 Quadratic Functions
                                  • 2.2.2 Factorising & Completing the Square
                                    • 2.2.3 Solving Quadratics
                                      • 2.2.4 Quadratic Inequalities
                                        • 2.2.5 Discriminants
                                        • 2.3 Functions Toolkit
                                          • 2.3.1 Language of Functions
                                            • 2.3.2 Composite & Inverse Functions
                                              • 2.3.3 Graphing Functions
                                              • 2.4 Further Functions & Graphs
                                                • 2.4.1 Reciprocal & Rational Functions
                                                  • 2.4.2 Exponential & Logarithmic Functions
                                                    • 2.4.3 Solving Equations
                                                      • 2.4.4 Modelling with Functions
                                                      • 2.5 Transformations of Graphs
                                                        • 2.5.1 Translations of Graphs
                                                          • 2.5.2 Reflections of Graphs
                                                            • 2.5.3 Stretches of Graphs
                                                              • 2.5.4 Composite Transformations of Graphs
                                                            • 3. Geometry & Trigonometry
                                                              • 3.1 Geometry Toolkit
                                                                • 3.1.1 Coordinate Geometry
                                                                  • 3.1.2 Radian Measure
                                                                    • 3.1.3 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 Further Trigonometry
                                                                                • 3.4.1 The Unit Circle
                                                                                  • 3.4.2 Exact Values
                                                                                  • 3.5 Trigonometric Functions & Graphs
                                                                                    • 3.5.1 Graphs of Trigonometric Functions
                                                                                      • 3.5.2 Transformations of Trigonometric Functions
                                                                                        • 3.5.3 Modelling with Trigonometric Functions
                                                                                        • 3.6 Trigonometric Equations & Identities
                                                                                          • 3.6.1 Simple Identities
                                                                                            • 3.6.2 Double Angle Formulae
                                                                                              • 3.6.3 Relationship Between Trigonometric Ratios
                                                                                                • 3.6.4 Linear Trigonometric Equations
                                                                                                  • 3.6.5 Quadratic Trigonometric Equations
                                                                                                • 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 Tranformations 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 & 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
                                                                                                                                        • 4.6.3 Standardisation of Normal Variables
                                                                                                                                      • 5. Calculus
                                                                                                                                        • 5.1 Differentiation
                                                                                                                                          • 5.1.1 Introduction to Differentiation
                                                                                                                                            • 5.1.2 Applications of Differentiation
                                                                                                                                            • 5.2 Further Differentiation
                                                                                                                                              • 5.2.1 Differentiating Special Functions
                                                                                                                                                • 5.2.2 Techniques of Differentiation
                                                                                                                                                  • 5.2.3 Second Order Derivatives
                                                                                                                                                    • 5.2.4 Further Applications of Differentiation
                                                                                                                                                      • 5.2.5 Concavity & Points of Inflection
                                                                                                                                                        • 5.2.6 Derivatives & Graphs
                                                                                                                                                        • 5.3 Integration
                                                                                                                                                          • 5.3.1 Introduction to Integration
                                                                                                                                                            • 5.3.2 Applications of Integration
                                                                                                                                                            • 5.4 Further Integration
                                                                                                                                                              • 5.4.1 Integrating Special Functions
                                                                                                                                                                • 5.4.2 Techniques of Integration
                                                                                                                                                                  • 5.4.3 Definite Integrals
                                                                                                                                                                    • 5.4.4 Further Applications of Integration
                                                                                                                                                                    • 5.5 Optimisation
                                                                                                                                                                      • 5.5.1 Modelling with Differentiation
                                                                                                                                                                      • 5.6 Kinematics
                                                                                                                                                                        • 5.6.1 Kinematics Toolkit
                                                                                                                                                                          • 5.6.2 Calculus for Kinematics
                                                                                                                                                                        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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