Edexcel A Level Further Maths: Core Pure:复习笔记3.2.2 Method of Differences

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  which sum to o
      • 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
    • 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  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 :

Exam Tip

  • Mark schemes often specify how many terms from the start and end of the series should be written down – it is usually two or three, so always write down the first three and last three terms
  • Don’t be afraid to write out more terms than this to make sure you spot the pattern, and can easily decide which terms will cancel and which will not
  • Check your algebraic answer by substituting in numbers to make sure it works; you can use your calculator to find summations in sigma notation

Worked Example

(a) Express  in partial fractions.




(b) Hence show that  using the method of differences.