IGCSEFurther MathsEdexcelRevision NotesSeriesArithmetic & Geometric SeriesArithmetic Sequences & Series

Arithmetic Sequences & Series (Edexcel IGCSE Further Pure Maths)

Revision Note

Roger B

Written by: Roger B

Reviewed by: Dan Finlay

Arithmetic Sequences

What is an arithmetic sequence?

  • In an arithmetic sequence, the difference between consecutive terms in the sequence is constant

    • This means a common difference is added to each term to get the next term

  • The first term of the sequence is denoted by a

  • The common difference is denoted by d 

    • For example, 1, 4, 7, 10, … is an arithmetic sequence with the rule ‘start at one and add three to each number’

      • The first term, bold italic a, is 1

      • The common difference, bold italic d, is 3

  • An arithmetic sequence can be

    • increasing (positive common difference), or

    • decreasing (negative common difference)

  • Terms in an arithmetic sequence can be referred to

    • by the letter u with

    • a subscript corresponding to its place in the sequence

      • e.g.  u subscript 1 equals a is the first term, u subscript 9 is the ninth term, u subscript n is the nth term, etc.

Arithmetic Series

What is an arithmetic series?

  • When the terms of an arithmetic sequence are added together, that is known as an arithmetic series

    • The terms (1st term, 2nd term, 3rd term, etc.) are exactly the same in the sequence and series

    • But with series we're most interested in what happens when the terms are added together

How do I find a term in an arithmetic series?

  • The nth term formula for an arithmetic sequence is

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

  • Where a is the first term, and d is the common difference

  • This is not given on the exam formula sheet, so make sure you know it

  • The formula allows you to find any term in the arithmetic series

    • Enter the values of a, r and n and calculate the value of u subscript n

  • Sometimes you will be given a term (bold italic u subscript bold n) and asked to find bold italic a or bold italic d

    • Substitute the information you have into the formula and solve the equation

  • Sometimes you will be given two terms and asked to find both bold italic a and bold italic d

    • Substitute the information into the formula and set up a pair of simultaneous equations

      • Then solve the simultaneous equations

How do I find the sum of an arithmetic series?

  • An arithmetic series is the sum of the terms in an arithmetic sequence

    • For the arithmetic sequence 1, 4, 7, 10, … the arithmetic series is 1 + 4 + 7 + 10 + …

  • Use the following formula to find the sum of the first n terms of an arithmetic series:

S subscript n equals n over 2 open square brackets 2 a plus space left parenthesis n minus 1 right parenthesis d close square brackets   

  • a is the first term

  • d is the common difference

  • The formula is given on the exam formula sheet

    • So you don't need to remember it

    • But you do need to know how to use it!

  • A question will often give you the sum of a certain number of terms and ask you to find the value of bold italic a or bold italic d

    • Substitute the information you have into the formula and solve the equation

Examiner Tips and Tricks

  • The formula for the sum of an arithmetic series is on the exam formula sheet

    • But the nth term formula is not on the formula sheet

  • Simultaneous equations are often needed within arithmetic series questions

    • Make sure you are confident solving them!

Worked Example

The fourth term of an arithmetic series is 10 and the ninth term is 25.  Find the first term and the common difference of the series.

Put the information for the fourth and ninth terms into the nth term formula  u subscript n equals a plus open parentheses n minus 1 close parentheses d

For u subscript 4

table row cell a plus open parentheses 4 minus 1 close parentheses d end cell equals 10 row cell a plus 3 d end cell equals 10 end table

For u subscript 9

table row cell a plus open parentheses 9 minus 1 close parentheses d end cell equals 25 row cell a plus 8 d end cell equals 25 end table

That gives us two simultaneous equations in a and d
Subtract the u subscript 4 equation from the u subscript 9 equation to eliminate a

table attributes columnalign right center left columnspacing 0px end attributes row cell 5 d end cell equals 15 row d equals 3 end table

Substitute that value into the first equation to find a

table row cell a plus 3 open parentheses 3 close parentheses end cell equals 10 row cell a plus 9 end cell equals 10 row a equals 1 end table

That is all the information we need to answer the question

bold italic a bold equals bold 1 bold comma bold space bold space bold space bold italic d bold equals bold 3

Worked Example

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

The first term is 18. 

a) Find the common difference, d, of the series.

Use the arithmetic series formula  S subscript n equals n over 2 open square brackets 2 a plus open parentheses n minus 1 close parentheses d close square brackets

Here  n equals 10,  S subscript 10 equals 630  and  a equals 18

Substitute in and solve for d

table row 630 equals cell 10 over 2 open square brackets 2 open parentheses 18 close parentheses plus open parentheses 10 minus 1 close parentheses d close square brackets end cell row 630 equals cell 5 open square brackets 36 plus 9 d close square brackets end cell row 630 equals cell 180 plus 45 d end cell row 450 equals cell 45 d end cell row d equals cell 450 divided by 45 end cell end table

bold italic d bold equals bold 10

The sum of the first 10 terms of another arithmetic series is also 630.

The common difference is 11. 

b) Find the first term, a, of the series.

Here  n equals 10,  S subscript 10 equals 630  and  d equals 11

table row 630 equals cell 10 over 2 open square brackets 2 a plus open parentheses 10 minus 1 close parentheses 11 close square brackets end cell row 630 equals cell 5 open square brackets 2 a plus 99 close square brackets end cell row 630 equals cell 10 a plus 495 end cell row 135 equals cell 10 a end cell row a equals cell 135 divided by 10 end cell end table

bold italic a bold equals bold 13 bold. bold 5

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    Author
    Reviewer
    Roger B

    Author: Roger B

    Expertise: Maths

    Roger's teaching experience stretches all the way back to 1992, and in that time he has taught students at all levels between Year 7 and university undergraduate. Having conducted and published postgraduate research into the mathematical theory behind quantum computing, he is more than confident in dealing with mathematics at any level the exam boards might throw at you.

    Dan Finlay

    Author: Dan Finlay

    Expertise: Maths Lead

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.