Prove that the sum of the reciprocals of the first n triangular numbers gets closer and closer to 2 as n grows.

Watch the video to see how Charlie works out the sum. Can you adapt his method?

Watch the video to see how to add together an arithmetic sequence of numbers efficiently.

Can you describe this route to infinity? Where will the arrows take you next?

Prove that if n is a triangular number then 8n+1 is a square number. Prove, conversely, that if 8n+1 is a square number then n is a triangular number.

Use algebra to reason why 16 and 32 are impossible to create as the sum of consecutive numbers.

Take ten sticks in heaps any way you like. Make a new heap using one from each of the heaps. By repeating that process could the arrangement 7 - 1 - 1 - 1 ever turn up, except by starting with it?