Sequence Investigation

This is a nice investigation idea from Nrich.  The above screen capture is from their Picture Story puzzle.  We have successive cubes – a 1x1x1 cube, a 2x2x2 cube etc.

The cubes are then rearranged to give the following shape.  The puzzle is then to use this information to discover a mathematical relationship.  This was my first attempt at this:

13 = 12
23 = (1+2)2 – 12
33 = (1+2+3)2 – (1+2)2
43 = (1+2+3+4)2 – (1+2+3)2

n3 = (1+2+3+4+…+n)2 – (1+2+3+…+ (n-1))2

This is not an especially attractive relationship – but nevertheless we have discovered a mathematical relationship using the geometrical figures above. Next let’s see why the RHS is the same as the LHS.

RHS:

(1+2+3+4+…+n)2 – (1+2+3+…+ (n-1))2

= ([1+2+3+4+…+ (n-1)] + n)2 – (1+2+3+…+ (n-1))2

= (1+2+3+…+ (n-1))2 + n2 + 2n(1+2+3+4+…+ (n-1)) – (1+2+3+…+ (n-1))2

= n2 + 2n(1+2+3+4+…+ (n-1))

next we notice that 1+2+3+4+…+ (n-1) is the sum of an arithmetic sequence first term 1, common difference 1 so we have:

1+2+3+4+…+ (n-1) = (n-1)/2 (1 + (n-1) )

1+2+3+4+…+ (n-1) = (n-1)/2 + (n-1)2/2

1+2+3+4+…+ (n-1) = (n-1)/2 + (n2 – 2n + 1)/2

Therefore:

2n(1+2+3+4+…+ (n-1)) = 2n ( (n-1)/2 + (n2 – 2n + 1)/2 )

2n(1+2+3+4+…+ (n-1)) = n2 -n + n3 – 2n2 + n

Therefore

n2 + 2n(1+2+3+4+…+ (n-1)) = n2 + n2 -n + n3 – 2n2 + n
n2 + 2n(1+2+3+4+…+ (n-1)) = n3

and we have shown that the RHS does indeed simplify to the LHS – as we would expect.

An alternative relationship

13 = 12
13+23 = (1+2)2
13+23+33  = (1+2+3)2
13+23+33+…n3  = (1+2+3+…+n)2

This looks a bit nicer – and this is a well known relationship between cubes and squares.  Could we prove this using induction?  Well we can show it’s true for n =1.  Then we can assume true for n=k:

13+23+33+…k3  = (1+2+3+…+k)2

Then we want to show true for n = k+1

ie.

13+23+33+… k3  + (k+1)3= (1+2+3+…+k + (k+1))2

LHS:

13+23+33+… k3  + (k+1)3

= (1+2+3+…+k)2 + (k+1)3

RHS:

(1+2+3+…+k + (k+1))2

= ([1+2+3+…+k] + (k+1) )2

= [1+2+3+…+k]2 + (k+1)2 + 2(k+1)[1+2+3+…+k]

= [1+2+3+…+k]+ (k+1)2 + 2(k+1)(k/2 (1+k))   (sum of a geometric formula)

= [1+2+3+…+k]+ (k+1)2 + 2(k+1)(k/2 (1+k))

= [1+2+3+…+k] + k3+ 3k2 + 3k + 1

= (1+2+3+…+k)2 + (k+1)3

Therefore we have shown that the LHS = RHS and using our induction steps have shown it’s true for all n.  (Write this more formally for a real proof question in IB!)

So there we go – a couple of different mathematical relationships derived from a simple geometric pattern – and been able to prove the second one (the first one would proceed in a similar manner).  This sort of free-style pattern investigation where you see what maths you can find in a pattern could make an interesting maths IA topic.

Essential resources for IB students:

1) Revision Village

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Revision Village has been put together to help IB students with topic revision both for during the course and for the end of Year 12 school exams and Year 13 final exams.  I would strongly recommend students use this as a resource during the course (not just for final revision in Y13!) There are specific resources for HL and SL students for both Analysis and Applications.

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There is a comprehensive Questionbank takes you to a breakdown of each main subject area (e.g. Algebra, Calculus etc) and then provides a large bank of graded questions.  What I like about this is that you are given a difficulty rating, as well as a mark scheme and also a worked video tutorial.  Really useful!

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The Practice Exams section takes you to a large number of ready made quizzes, exams and predicted papers.   These all have worked solutions and allow you to focus on specific topics or start general revision.  This also has some excellent challenging questions for those students aiming for 6s and 7s.

Each course also has a dedicated video tutorial section which provides 5-15 minute tutorial videos on every single syllabus part – handily sorted into topic categories.

2) Exploration Guides and Paper 3 Resources

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I’ve put together four comprehensive pdf guides to help students prepare for their exploration coursework and Paper 3 investigations. The exploration guides talk through the marking criteria, common student mistakes, excellent ideas for explorations, technology advice, modeling methods and a variety of statistical techniques with detailed explanations. I’ve also made 17 full investigation questions which are also excellent starting points for explorations.  The Exploration Guides can be downloaded here and the Paper 3 Questions can be downloaded here.