2025 SAJC P2 Q4

A sequence \(u_1, u_2, u_3, \dots\) is such that \(23u_{r+1} = 19u_r + 16\), for \(r > 0\) and \(u_1 = 2\).

1. Write down the value of \(u_{10}\), giving your answer correct to \(4\) decimal places.[1]
2. It is given that as \(r \to \infty\), \(u_r \to l\). Show that \(l = 4\).[1]
3. Hence, find the smallest integer \(r\) for which \(u_r\) exceeds \(99.9\)% of \(l\).[3]

It is known that the \(k\)th term of this sequence is given by \(u_k = 4 - 2\left(\frac{19}{23}\right)^{k-1}\).

4. Given that a new series \(S_n\) is defined by \(S_n = \sum_{k=1}^{n} (u_k - c)\), where \(c\) is a real constant. Express \(S_n\) in the form of \(An + B\left[1 - \left(\frac{19}{23}\right)^n\right]\), where \(A\) is a constant in terms of \(c\) and \(B\) is a real number.[3]
5. Determine the possible values of \(c\) for \(S_n\) to diverge as \(n \to \infty\).[1]