A sequence of integers \(u_n\) is generated by the relation \(u_{n+1} = u_n + u_{n-1}\). Show that the sequence of remainders when the \(u_n\) are divided by a fixed integer \(k\) is periodic. Deduce that if \(u_0 = -1\) and \(u_1 = 1\) then some \(u_n\) is divisible by \(k\). By considering the case \(k = 5\), show that this last result is not true for all pairs of initial values \(u_0\) and \(u_1\).
The function \(f\) is differentiable and satisfies the identity \[ f(x) + f(y) = f\left(\frac{xy}{x+y+1}\right) \] for \(x, y > 0\). Show that \(x(x+1)f'(x)\) is constant, and hence deduce the function \(f\).