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IBMathsDPApplications & Interpretation (AI)SLRevision Notes1. Number & AlgebraSequences & SeriesGeometric Sequences & Series

Geometric Sequences & Series (DP IB Applications & Interpretation (AI)): Revision Note

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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

 

Examiner Tips and Tricks

  • 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

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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

Examiner Tips and Tricks

  • 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

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    Amber

    Author: Amber

    Expertise: Maths

    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.