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Poly Fibs

A sequence of polynomials starts 0, 1 and each poly is given by combining the two polys in the sequence just before it. Investigate and prove results about the roots of the polys.

Fixing It

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?

OK! Now Prove It

Make a conjecture about the sum of the squares of the odd positive integers. Can you prove it?

Fibonacci Factors

Age 16 to 18 Challenge Level:

Image of a yellow shell

In the Fibonacci sequence each term is the sum of the two terms before it:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144...

Where do the even numbers come in the sequence?

Is there a pattern? Why?

Which Fibonacci numbers are divisible by 3?

Prove general results for the occurrence of even Fibonacci numbers in the sequence and for the occurrence of multiples of 3.