### 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:

Denoting the Fibonacci numbers $1,1,2,3,5,8,...$ by $f_n$ where $f_n=f_{n-1}+ f_{n-2}$ prove for all positive integer values of $n$ that $\frac{f_n}{f_{n+2}}$ and $\frac{f_{n+1}}{f_{n+3}}$ are Farey neighbours, that is $|f_{n+1}f_{n+2}-f_nf_{n+3}|=1$.

Show that the mediant of $\frac{f_n}{f_{n+2}}$ and $\frac{f_{n+1}}{f_{n+3}}$ is $\frac{f_{n+2}}{f_{n+4}}$.

See the problem Farey Neighbours.