Converging Product

In the limit you get the sum of an infinite geometric series. What about an infinite product (1+x)(1+x^2)(1+x^4)... ?

OK! Now Prove It

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

Overarch 2

Bricks are 20cm long and 10cm high. How high could an arch be built without mortar on a flat horizontal surface, to overhang by 1 metre? How big an overhang is it possible to make like this?

Farey Fibonacci

Stage: 5 Short Challenge Level:

Test the result for several small values of $n$ and try to decide why the absolute value appears in the formula.

What values of $n$ are you asked to prove the result for and what method of proof does this suggest?