What is an arithmetic sequence or arithmetic series?

• Ensure you are familiar with Sequences – Basics and Linear

• 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 seriesSome 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 numbers1 2 3 4 5 6 7 8 9 10
  • Write them backwards10 9 8 7 6 5 4 3 2 1
  • Add up both lists11? 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