Past fortnightly questions

This question is about rewriting functions in a different form.  No knowledge beyond GCSE is needed to complete the question.

The original STEP question had no question parts.  We have included some to make it clearer how the different requests in the question are related to each other.

This question involves substituting numbers into functions, simultaneous equations, and a bit of very basic number theory - for example, if $p$ and $q$ are both integers then so is $p+q$.

If you have not met it already, the formula for the triangular numbers is $\frac{n(n+1)}2$.  This is always an integer as we know that the triangular numbers are integers. Alternatively you could argue that $n$ and $n+1$ are consecutive numbers so one of them is even, and so has a factor of two.  Hence $\frac{n(n+1)}2$ must be an integer.