Non-linear or complex iterative formula convergence

3 The sequence of values given by the iterative formula $$x _ { n + 1 } = \frac { 3 x _ { n } } { 4 } + \frac { 15 } { x _ { n } ^ { 3 } }$$ with initial value \(x _ { 1 } = 3\), converges to \(\alpha\).

1. Use this iterative formula to find \(\alpha\) correct to 2 decimal places, giving the result of each iteration to 4 decimal places.
2. State an equation satisfied by \(\alpha\) and hence find the exact value of \(\alpha\).

8 A sequence \(\left\{ \mathrm { u } _ { \mathrm { n } } \right\}\) is defined by the recurrence system \(u _ { 1 } = 1\) and \(\mathrm { u } _ { \mathrm { n } + 1 } = \mathrm { a } - \frac { \mathrm { a } ^ { 2 } } { 2 \mathrm { u } _ { \mathrm { n } } }\) for \(n \geqslant 1\), where \(a\) is a positive constant.
Determine with justification the behaviour of the sequence for all possible values of \(a\). \section*{END OF QUESTION PAPER}

11 A sequence of terms \(x _ { n }\) generated by a recurrence relation is said to be strictly increasing if, for each \(x _ { n } , x _ { n + 1 } > x _ { n }\).

1. Let a recurrence relation be defined by $$x _ { n + 1 } = \frac { x _ { n } ^ { 2 } + 2 } { 3 } \quad \text { and } \quad x _ { 0 } = \frac { 1 } { 2 } \quad \text { for } n \geq 0$$ Calculate \(x _ { 1 } , x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\) correct to 3 significant figures where appropriate.
2. Given the recurrence relation $$x _ { n + 1 } = \frac { x _ { n } ^ { 2 } + 2 } { 3 }$$ show that the sequence is strictly increasing when \(x _ { n } > 2\) or \(x _ { n } < 1\).
3. If \(- 1 < x _ { 0 } < 1\), then the sequence \(x _ { n } ( n \geq 0 )\) converges to a limit. Explain briefly why this limit is 1 .
4. Given the recurrence relation $$x _ { n + 1 } = \frac { x _ { n } ^ { 2 } + k } { m } \text { with } m > 0$$ prove that \(x _ { n }\) is a strictly increasing sequence for all \(x _ { n }\) if \(m ^ { 2 } < 4 k\).

The sequence \(u_1, u_2, u_3, \ldots\) is such that \(u_1 = 1\) and $$u_{n+1} = -1 + \sqrt{(u_n + 7)}.$$

1. Prove by induction that \(u_n < 2\) for all \(n \geqslant 1\). [4]
2. Show that if \(u_n = 2 - \varepsilon\), where \(\varepsilon\) is small, then $$u_{n+1} \approx 2 - \frac{1}{6}\varepsilon.$$ [2]

The sequence of positive numbers \(u_1\), \(u_2\), \(u_3\), \(\ldots\) is such that \(u_1 < 3\) and, for \(n \geqslant 1\), $$u_{n+1} = \frac{4u_n + 9}{u_n + 4}.$$

1. By considering \(3 - u_{n+1}\), or otherwise, prove by mathematical induction that \(u_n < 3\) for all positive integers \(n\). [5]
2. Show that \(u_{n+1} > u_n\) for \(n \geqslant 1\). [3]