Arithmetic Sequences & Series (Edexcel IGCSE Maths)

What is an arithmetic sequence or arithmetic series?

• An arithmetic sequence is a sequence of numbers that increase or decrease by the same amount from one term to the next
  • This amount is called the common difference
  • eg. 5, 9, 13, 17, 21, ...  common difference of 4
  • eg2. 24, 17, 10, 3, -4, ..., -95  common difference of -7

• An arithmetic series is where terms are added together
  • eg. 5 + 9 + 13 + 17 + 21 + ...
  • eg2. 24 + 17 + 10 + 3 + -4 + ... + -95

• Lots of letters are used in sequences, make sure you are familiar with them
  • a – the first term in an arithmetic series
  • d – the common difference of an arithmetic series
  • n – the number of terms in the arithmetic series
    Some series go on forever (ie.  have an infinite number of terms – but it could be that only the first 10 terms, say, are of interest, so n = 10)

• S_n is used for the sum of the first n terms of an arithmetic series

How do I find the sum of an arithmetic series?

• There is a formula for adding up the first n terms of an arithmetic series
  • The formula is included on the formulae sheet

• You do not need to where the formula comes from but, just for fun, here’s a hint
  • To add up the numbers 1 to 10
  • Write out the numbers

    1 2 3 4 5 6 7 8 9 10

  • Write them backwards

    10 9 8 7 6 5 4 3 2 1

  • Add up both lists

    11  11  11  11  11  11  11  11  11 11

  • This is 10 × 11 = 110
  • But this is twice the sum as two lots were added together
  • So the sum of the numbers 1 to 10 is 110 ÷ 2 = 55