Let a(n) be the number of ways of expressing the integer n as an
ordered sum of 1's and 2's. Let b(n) be the number of ways of
expressing n as an ordered sum of integers greater than 1. (i)
Calculate a(n) and b(n) for n<8. What do you notice about these
sequences? (ii) Find a relation between a(p) and b(q). (iii) Prove
your conjectures.

Can you find a rule which connects consecutive triangular numbers?

In Particular

Stage: 4 Challenge Level:

Write 100 as the sum of two positive integers, one divisible by
7 and the other divisible by 11. Use your answer to find formulas
giving all the solutions of the following equation where x and y
are integers.