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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.

Lower Bound

Age 14 to 16
Challenge Level

Younger learners can work with numbers and find out what happens to the sequence.

As you work out more terms does the value seem to get closer to a particular number? Are the values always greater than that number? Why do you think this happens?

If you want to go on to use algebra, a bit of algebra will help you to explain what happens and to prove that there is a limiting value for this sequence. However you will have to know how to multiply two bracketed terms together to do this so it might be something to come back to when you have learnt a little more algebra.