Arithmetic Series (CIE A Level Maths: Pure 1)

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Arithmetic Series

How do I find the sum of an arithmetic progression?

  • The sum of the terms of an arithmetic progression is sometimes called an arithmetic series

Arithm Series Illustr, A Level & AS Level Pure Maths Revision Notes

  • The following formulae will let you find the sum of the first n terms of an arithmetic progression:
begin mathsize 22px style S subscript n equals n over 2 left parenthesis 2 a plus left parenthesis n minus 1 right parenthesis d right parenthesis end style   or   begin mathsize 22px style S subscript n equals n over 2 left parenthesis a plus l right parenthesis end style
    • is the first term
    • is the common difference
    • is the last term

  • You can use whichever formula is more convenient for a given question
  • The a and the d in those formulae are exactly the same as the ones used with arithmetic progressions

 

How do I derive the formula for the sum of an arithmetic progression?

  • Learn this proof of the sum of an arithmetic progression formula – you can be asked to give it on the exam:
    • Write the terms out once in order
    • Write the terms out again in reverse order
    • Add the two sums together
      • The terms will pair up to give the same sum 2 a plus left parenthesis n minus 1 right parenthesis d
      • There will be of these terms
    • Divide by two as two of the sums have been added togetherArithm Series Proof, A Level & AS Level Pure Maths Revision Notes

Examiner Tip

The arithmetic series formulae are in the formulae booklet – you don't need to memorise them.

Worked example

4-2-2-arithm-series-example

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Roger

Author: Roger

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.