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'Particularly General' printed from https://nrich.maths.org/
Why do this problem?
This problem gives practice and insight into manipulating and constructing algebraic identities. It is based upon the idea of 'proof by generic example': the proof method of a particular example encapsulates in a clear way all of the key points of a proof of a more general case. Manipulation of algebraic identities is always good fun and will give very solid general preparation for the more
challenging end of school mathematics examinations, such as Further Mathematics and STEP. It can also be used to help students to understand the differences in the uses of direct proof and proof by induction.
Possible approach
This problem is well suited to individual solving and will require a reasonable fluency in algebraic manipulation. It would most naturally accompany work on proof by induction.
A brief discussion on proof could follow attempts at this question if you feel comfortable with the topic. Some discussion points of interest are:
o Proof by induction can often be used to prove identities involving a variable $n$, but frequently provide no insight as to why the identities make sense, or where they came from in the first place.
o These algebraic proofs by generic example indicate clearly WHY the results are true, but for large $n$ the actual algebraic proofs would be very long sequences of symbolic manipulations. Do we need to write these down?
o In some sense, the generalisations are justified by understanding that the manipulations made will, in some way, continue. How do students feel about this lack of formality?
o Remind students that working through the algebra in specific instances still constitutes a 'direct' proof!
Key questions
What is the difference between an identity and an equation?
What method of proof can you use for the specific examples given?
How can you expand the brackets or reduce the terms in a step by step way? Can you see how this procedure might generalise?
Possible extension
Using the ideas gained from solving this problem, can students construct some new identities of their own?
Possible support
Suggest the use of difference of two squares and a trigonometric double angle formula.