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}.

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.

Screen Shot

Age 14 to 16 Challenge Level:

This problem gives opportunities to address processes that need splitting into parts.

The path of the light ray switches direction at intervals, and handling that is crucial to the problem.

Mathematical reasoning & proof. Expanding and factorising quadratics. Inequalities. Sine, cosine, tangent. Creating and manipulating expressions and formulae. Generalising. Reflections. Regular polygons and circles. Simultaneous equations. Polynomial functions and their roots.

You may also like

Diophantine N-tuples

DOTS Division

Sixational

Screen Shot

Age 14 to 16 Challenge Level: