A sequence is such that $u_1=p$, where $p$ is a constant, and \(u_{n+1}=\frac{6u_n}{4u

Question

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]

Accepted 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 $3k$.

Nonlinear Recurrence and Arithmetic Differences

Nonlinear Recurrence and Arithmetic Differences