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IBMathsDPAnalysis & Approaches (AA)SLRevision Notes1. Number & AlgebraSequences & SeriesArithmetic Sequences & Series

Arithmetic Sequences & Series (DP IB Analysis & Approaches (AA)): Revision Note

Amber

Written by: Amber

Reviewed by: Mark Curtis

Updated on

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

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Arithmetic Sequences

What is an arithmetic sequence?

  • An arithmetic sequence is a sequence in which the difference between consecutive terms is constant

    • This constant difference is known as the common difference, d

      • e.g. for the sequence 1, 4, 7, 10, … d equals 3

  • A negative common difference means the sequence decreases

How do I find a term in an arithmetic sequence?

  • The nth term formula for an arithmetic sequence is

u subscript n equals u subscript 1 plus left parenthesis n minus 1 right parenthesis d

  • Where u subscript 1 is the first term, and d is the common difference

Examiner Tips and Tricks

The formula for the nth term in an arithmetic sequence is given in the formula booklet.

Examiner Tips and Tricks

In harder arithmetic sequence questions, it is common to have to solve simultaneous equations in u subscript 1 and d.

Worked Example

The fourth term of an arithmetic sequence is 10 and the ninth term is 25.

Find the first term and the common difference of the sequence.

ai-sl-1-2-2-arit-sequences

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Arithmetic Series

How do I find the sum of an arithmetic sequence?

  • The formulae to find the sum of the first n terms of an arithmetic sequence, S subscript n, are

table row cell S subscript n end cell equals cell n over 2 left parenthesis 2 u subscript 1 plus left parenthesis n minus 1 right parenthesis d right parenthesis end cell row cell S subscript n end cell equals cell n over 2 left parenthesis u subscript 1 plus u subscript n right parenthesis end cell end table   

  • Where

    • u subscript 1 is the first term

    • d is the common difference

    • u subscript n is the last term

Examiner Tips and Tricks

Both formulae for the sum of the first n terms in an arithmetic sequence are given in the formula booklet (use which ever one is more convenient for the question).

Examiner Tips and Tricks

In harder questions on arithmetic series, it is common to have to solve quadratics in n or simultaneous equations in u subscript 1 and d.

Worked Example

The sum of the first 10 terms of an arithmetic sequence is 630.

(a) If the first term is 18, find the common difference, d.

ai-sl-1-2-2-arit-seriesa

(b) If, instead, the common difference is 11, find the first term of the sequence.

ai-sl-1-2-2-arit-seriesb

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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.