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Diophantine N-tuples

Can you explain why a sequence of operations always gives you perfect squares?

DOTS Division

Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.

Sixational

The nth term of a sequence is given by the formula n^3 + 11n . Find the first four terms of the sequence given by this formula and the first term of the sequence which is bigger than one million. Prove that all terms of the sequence are divisible by 6.

Triangles Within Pentagons

Age 14 to 16 Challenge Level:

The diagram shows the groupings of the numbers which is mirrored in the derivation of the formulae.
The rule is generalisable but can you convince us why?

For the last part you need a formula for triangular numbers. Each triangular number is the sum of all the whole numbers so the fifth triangular number is 1+2+3+4+5.

By reversing and adding any group of consecutive numbers to itself you can generate the triangular numbers. The thesuarus might help here.

You might find it helpful to visualise the pentagonal numbers as made from triangular numbers.

Now you should notice that the formula for the pentagon numbers can be written in terms of triangular numbers.

There is a little bit of agebraic substitution necessary to get you to the point where you can show that every pentagonal number is a third of a triangular number.

Is the inverse the case?