Problems

The sequence of real numbers \(u_1\), \(u_2\), \(u_3\), \(\ldots\) is defined by \begin{equation*} u_1=2 \,, \qquad\text{and} \qquad u_{n+1} = k - \frac{36}{u_n} \quad \text{for } n\ge1, \tag{\(*\)} \end{equation*} where \(k\) is a constant.

1. Determine the values of \(k\) for which the sequence \((*)\) is: (a) constant; (b) periodic with period 2; (c) periodic with period 4.
2. In the case \(k=37\), show that \(u_n\ge 2\) for all \(n\). Given that in this case the sequence \((*)\) converges to a limit \(\ell\), find the value of \(\ell\).

The sequence \(u_n\) (\(n= 1, 2, \ldots\)) satisfies the recurrence relation \[ u_{n+2}= \frac{u_{n+1}}{u_n}(ku_n-u_{n+1}) \] where \(k\) is a constant. If \(u_1=a\) and \(u_2=b\,\), where \(a\) and \(b\) are non-zero and \(b \ne ka\,\), prove by induction that \[ u_{2n}=\Big(\frac b a \Big) u_{2n-1} \] \[ u_{2n+1}= c u_{2n} \] for \(n \ge 1\), where \(c\) is a constant to be found in terms of \(k\), \(a\) and \(b\). Hence express \(u_{2n}\) and \(u_{2n-1}\) in terms of \(a\), \(b\), \(c\) and \(n\). Find conditions on \(a\), \(b\) and \(k\) in the three cases:

1. the sequence \(u_n\) is geometric;
2. \(u_n\) has period 2;
3. the sequence \(u_n\) has period 4.

Given a sequence \(w_0\), \(w_1\), \(w_2\), \(\ldots\,\), the sequence \(F_1\), \(F_2\), \(\ldots\) is defined by $$F_n = w_n^2 + w_{n-1}^2 - 4w_nw_{n-1} \,.$$ Show that $\; F_{n}-F_{n-1} = \l w_n-w_{n-2} \r \l w_n+w_{n-2}-4w_{n-1} \r \; \( for \)n \ge 2\,$.

1. The sequence \(u_0\), \(u_1\), \(u_2\), \(\ldots\) has \(u_0 = 1\), and \(u_1 = 2\) and satisfies \[ u_n = 4u_{n-1} -u_{n-2} \quad (n \ge 2)\;. \] Prove that \ $ u_n^2 + u_{n-1}^2 = 4u_nu_{n-1}-3 \; $ for \(n \ge 1\,\).
2. A sequence \(v_0\), \(v_1\), \(v_2\), \(\ldots\,\) has \(v_0=1\) and satisfies \begin{equation*} v_n^2 + v_{n-1}^2 = 4v_nv_{n-1}-3 \quad (n \ge 1). \tag{\(\ast\)} \end{equation*} \makebox[7mm]{(a) \hfill}Find \(v_1\) and prove that, for each \(n\ge2\,\), either \(v_n= 4v_{n-1} -v_{n-2}\) or \(v_n=v_{n-2}\,\). \makebox[7mm]{(b) \hfill}Show that the sequence, with period 2, defined by \begin{equation*} v_n = \begin{cases} 1 & \mbox{for \(n\) even} \\ 2 & \mbox{for \(n\) odd} \end{cases} \end{equation*} \makebox[7mm]{\hfill}satisfies \((\ast)\). \makebox[7mm]{(c) \hfill}Find a sequence \(v_n\) with period 4 which has \(v_0=1\,\), and satisfies~\((\ast)\).

A curve is defined parametrically by \[ x=t^2 \;, \ \ \ y=t (1 + t^2 ) \;. \] The tangent at the point with parameter \(t\), where \(t\ne0\,\), meets the curve again at the point with parameter \(T\), where \(T\ne t\,\). Show that \[ T = \frac{1 - t^2 }{2t} \mbox { \ \ \ and \ \ \ } 3t^2\ne 1\;. \] Given a point \(P_0\,\) on the curve, with parameter \(t_0\,\), a sequence of points \(P_0 \, , \; P_1 \, , \; P_2 \, , \ldots\) on the curve is constructed such that the tangent at \(P_i\) meets the curve again at \(P_{i+1}\). If \(t_0 = \tan \frac{ 7 } {18}\pi\,\), show that \(P_3 = P_0\) but \(P_1\ne P_0\,\). Find a second value of \(t_0\,\), with \(t_0>0\,\), for which \(P_3 = P_0\) but \(P_1\ne P_0\,\).