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
Can you find a rule which connects consecutive triangular numbers?
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
The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the
NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to
embed rich mathematical tasks into everyday classroom practice. More information on many of our other activities
can be found here.