A sequence is such that \(u_1=p\), where \(p\) is a constant, and \(u_{n+1}=\frac{6u_n}{4u_n+1}\) for \(n\geq1\).
Describe how the sequence behaves when \(p=1\).[2]
Find the value of \(p\) for which \(u_5=\frac{864}{691}\).[2]
Another sequence \(v_1,v_2,v_3,\ldots\) is such that, for all \(n\geq1\), \(v_{n+2}-2v_{n+1}+v_n=3k\), where \(k\) is a constant. Let \(w_n=v_{n+1}-v_n\) for \(n\geq1\). Explain why the sequence \(\{w_n\}\) is an arithmetic progression, stating its common difference.[2]
Similar questions are unavailable for this question.
Answer:(a)(i) The sequence increases and converges to \(\frac54\). (a)(ii) \(p=2\). (b) \(w_{n+1}-w_n=3k\), so \(\{w_n\}\) is an arithmetic progression with common difference \(3k\).