Revision NotesExam QuestionsPast Papers
O LevelAdditional MathsCambridgeRevision NotesSequences & SeriesArithmetic & Geometric ProgressionsArithmetic Progressions

Arithmetic Progressions (Cambridge O Level Additional Maths)

Revision Note

Amber

Author

Amber

Last updated

Arithmetic Sequences

What is an arithmetic progression?

  • In an arithmetic progression (also called arithmetic sequence), the difference between consecutive terms in the sequence is constant
  • This constant difference is known as the common difference, d, of the sequence
  • 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, a, is 1
    • The common difference, d, is 3
  • An arithmetic progression can be increasing (positive common difference) or decreasing (negative common difference)
  • Each term of an arithmetic progression is referred to by the letter u with a subscript determining its place in the sequence

Examples of arithmetic progressions

How do I find a term in an arithmetic progression?

  • The nth term formula for an arithmetic progression is given as

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 given on the list of formulas page of the exam, you do not need to know how to derive it
  • Sometimes you will be given a term and asked to find the first term or the common difference
    • Substitute the information into the formula and solve the equation
  • Sometimes you will be given two terms and asked to find both the first term and the common difference
    • Substitute the information into the formula and set up two simultaneous equations
    • Solve the simultaneous equations

Examiner Tip

  • Simultaneous equations are often needed within arithmetic progression questions
    • Make sure you are confident solving them

Worked example

Arithm Seq Example, A Level & AS Level Pure Maths Revision Notes

Arithmetic Series

What is an arithmetic series?

  • An arithmetic series is the sum of the terms in an arithmetic progression
    • It is often referred to as the sum of an arithmetic progression
    • For the arithmetic sequence 1, 4, 7, 10, … the arithmetic series is 1 + 4 + 7 + 10 + …

How do I find the sum of an arithmetic progression?

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

S subscript n equals 1 half n open curly brackets a space plus space l close curly brackets space equals 1 half n open curly brackets 2 a plus space left parenthesis n minus 1 right parenthesis d close curly brackets space   

      • a is the first term
      • l is the last term
      • d is the common difference
      • n is the number of terms in the series
    • Both formulae are given on the formula page, you do not need to know how to derive them
  • You can use whichever formula is more convenient for a given question
    • If you know the first term and common difference use the second version
    • If you know the first and last term then the first version is easier to use
  • 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 or the common difference
    • Substitute the information into the formula and solve the equation

Examiner Tip

  • The arithmetic series formulae are given on page 2 of the exam paper – you don't need to memorise them
    • Practise finding the formulae so that you can quickly locate them in the exam

Worked example

4-2-2-arithm-series-example

You've read 0 of your 5 free revision notes this week

Sign up now. It’s free!

Join the 100,000+ Students that ❤️ Save My Exams

the (exam) results speak for themselves:

    Test yourself

    Did this page help you?

    Previous:Language of Sequences & SeriesNext:Geometric Progressions
    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.