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
Stage: 4 Challenge Level:
This problem gives opportunities to address processes that need
splitting into parts.
The path of the light ray switches direction at intervals, and
handling that is crucial to the problem.
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.