A and B are two fixed points on a circle and RS is a variable diamater. What is the locus of the intersection P of AR and BS?
Two semicircle sit on the diameter of a semicircle centre O of
twice their radius. Lines through O divide the perimeter into two
parts. What can you say about the lengths of these two parts?
Make a conjecture about the sum of the squares of the odd positive integers. Can you prove it?
The Fibonnaci sequence occurs so frequently because it is the
solution of the simplest of all difference relations. It is
instructive to view it in this way and perhaps to introduce the
idea of difference equations with this familiar example.
Proving these results calls for considering whether or not other
terms in the sequences, apart from those in the recognized
patterns, can also be multiples of 2 or 3 respectively in the two
cases. Are the conditions necessary as well as sufficient?