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

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

Amber

Written by: Amber

Reviewed by: Mark Curtis

Updated on

DP Applications & Interpretation (AI): SLDP Applications & Interpretation (AI): HLDP Analysis & Approaches (AA): HLDP Analysis & Approaches (AA): SL

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Geometric Sequences

What is a geometric sequence?

  • A geometric sequence is a sequence where you multiply by a fixed number each time to get the next term in the sequence

    • The fixed number that you multiply by is called the common ratio, r

      • e.g. for 2, 6, 18, 54, 162, … r equals 3

  • A geometric sequence can be

    • increasing if r greater than 1

    • decreasing if 0 less than r less than 1

    • alternating if r less than 0

      • changing between positive and negative

How do I find a term in a geometric sequence?

  • The nth term formula for a geometric sequence is

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

Examiner Tips and Tricks

The formula for the nth term of a geometric sequence is given in the formula booklet.

Examiner Tips and Tricks

Some good tricks for geometric sequences are

  • dividing two terms by each other to eliminate u subscript 1 when finding r

  • using logarithms when finding n

Worked Example

The sixth term of a geometric sequence is 486 and the seventh term is 1458. 

Find

(a) the common ratio of the sequence,

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

(b) the first term of the sequence.

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

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Geometric Series

How do I find the sum of a geometric sequence?

  • The formulae for the sum of the first n terms of a geometric sequence are:

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

  • Where

    • u subscript 1 is the first term

    • r is the common ratio

Examiner Tips and Tricks

Both formulae for the sum of the first n terms of a geometric sequence are given in the formula booklet (the first one is easier when r space greater than space 1 and the second when r space less than space 1).

Examiner Tips and Tricks

Harder questions on geometric series may require the use of logarithms.

Worked Example

A geometric sequence has a first term of 25 and a common ratio of 0.8.

Find the fifth term and the sum of the first five terms.

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

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    Author
    Reviewer
    Amber

    Author: Amber

    Expertise: Maths Content Creator

    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.

    Mark Curtis

    Reviewer: Mark Curtis

    Expertise: Maths Content Creator

    Mark graduated twice from the University of Oxford: once in 2009 with a First in Mathematics, then again in 2013 with a PhD (DPhil) in Mathematics. He has had nine successful years as a secondary school teacher, specialising in A-Level Further Maths and running extension classes for Oxbridge Maths applicants. Alongside his teaching, he has written five internal textbooks, introduced new spiralling school curriculums and trained other Maths teachers through outreach programmes.