Method of Differences (Edexcel A Level Further Maths): Revision Note

Method of Differences

What is the Method of Differences?

• The Method of Differences is a way of turning longer and more complicated sums into shorter and simpler ones

• Sometimes when summing series, you will notice that many of the terms (or parts of the terms) simply “cancel out” or eliminate each other

  • This can turn a very long series summation, into a much simpler shorter one

• In the case of 

  • 

  • 

  • 

  • 

  • (and so on, until…)

  • 

    • This is the penultimate term

  • 

    • This is the last term

• You can see that when these are summed, most of the terms will cancel out

  • This leaves just 

  • We can say that 

How can I use partial fractions along with the method of differences?

• You will often need to use partial fractions to change the general term into a sum of two or three terms, rather than a single fraction

  • For example, can be rewritten as  

• This may lead to a more interesting pattern of cancellations than was seen for 

• For example,  can be written as , where , and the terms can then be listed as:   

  • 

    • 

    • 

    • 

    • 

    • (and so on, until…)

    • 

    • 

  • When these are summed, it will just leave 

  • You can then evaluate this expression with to get to your final answer

• It is helpful to use  notation to spot the pattern, rather than substituting into the expression every time, especially with more complicated expressions

  • You need to consider carefully which term to make  and then how the other terms in the expression relate to it

  • If this is difficult for a particular expression, it may be more straightforward to substitute  into each term in the series and spot any patterns that way 

  • This is essentially writing the series out in full until you spot which terms will cancel

  • In your working, however, you should still write out the last two or three terms in terms of , etc.

How can I use the method of differences for series with expressions containing more than two terms?

• The general term of the series may have more than two terms, which can sometimes make spotting which terms will cancel more challenging

  • For example,  

  • This can be written as where 

  • Writing out the first five terms and the last three terms we get:

    • 

    • 

    • 

    • 

    • 

    • (and so on until…)

    • 

    • 

    • 

  • In this case, look at the diagonals starting at the top right

    • We have and which sum to 

    • This pattern repeats for the other diagonals

  • We will eventually be left with only

  • Evaluating this with  results in an answer of  

What other uses are there for the method of differences?

• Method of differences can also be used to prove the formulae for the sum of and

  • For example, this can be used to prove that the result for the sum of squares is indeed 

    • By expanding brackets it can be shown that ,  and then the sum of both sides of that equation from to can be considered

    • 

    • The left-hand side can be found in terms of using method of differences, and the right-hand side can be rearranged to give 

    • That equation can then be rearranged to give an expression for 

    • When these proofs have appeared previously in exams, they have tended to be structured to help you work through the steps

• You may have to use your algebraic method of differences result to find a numerical answer, usually in the last part of a question

  • The question will often ask you to evaluate the sum starting from  (or some other arbitrary value) rather than from

  • To help with this, remember that: 

  • You may also find it helpful to recall that for constants and :