The nth term of a sequence is given by the formula n^3 + 11n . Find
the first four terms of the sequence given by this formula and the
first term of the sequence which is bigger than one million. Prove
that all terms of the sequence are divisible by 6.

Investigate sequences given by $a_n = \frac{1+a_{n-1}}{a_{n-2}}$
for different choices of the first two terms. Make a conjecture
about the behaviour of these sequences. Can you prove your
conjecture?

Three Frogs

Stage: 4 Challenge Level:

What possible positions are there with two frogs and after how many hops
is each possible - this might give a clue to the solution.

How many different positions are there and how do they relate to each
other?