Sequence Behaviour Classification

7 For each of the following sequences, write down sufficient terms of the sequence in order to be able to describe its behaviour as divergent, periodic or convergent. For any convergent sequence, state its limit.

1. \(a _ { 1 } = - 1 ; \quad a _ { k + 1 } = \frac { 4 } { a _ { k } }\)
2. \(\quad a _ { 1 } = 1 ; \quad a _ { k } = 2 - 2 \times \left( \frac { 1 } { 2 } \right) ^ { k }\)
3. \(\quad a _ { 1 } = 0 \quad a _ { k + 1 } = \left( 1 + a _ { k } \right) ^ { 2 }\).

1 Three sequences, \(\mathrm { a } _ { \mathrm { n } } , \mathrm { b } _ { \mathrm { n } }\) and \(\mathrm { c } _ { \mathrm { n } }\), are defined for \(n \geqslant 1\) by the following recurrence relations. $$\begin{aligned} & \left( a _ { n + 1 } - 2 \right) \left( 2 - a _ { n } \right) = 3 \text { with } a _ { 1 } = 3 \\ & b _ { n + 1 } = - \frac { 1 } { 2 } b _ { n } + 3 \text { with } b _ { 1 } = 1.5 \\ & c _ { n + 1 } - \frac { c _ { n } ^ { 2 } } { n } = 1 \text { with } c _ { 1 } = 2.5 \end{aligned}$$ The output from a spreadsheet which presents the first 10 terms of \(a _ { n } , b _ { n }\) and \(c _ { n }\), is shown below. Without attempting to solve any recurrence relations, describe the apparent behaviour, including as \(n \rightarrow \infty\), of

• \(a _ { n }\)
• \(\mathrm { b } _ { \mathrm { n } }\)
• \(\mathrm { C } _ { \mathrm { n } }\)

6 For real constants \(a\) and \(b\), the sequence \(U _ { 1 } , U _ { 2 } , U _ { 3 } , \ldots\) is given by $$U _ { 1 } = a \text { and } U _ { n } = \left( U _ { n - 1 } \right) ^ { 2 } - b \text { for } n \geqslant 2 .$$

1. Determine the behaviour of the sequence in the case where \(a = 1\) and \(b = 3\).
2. In the case where \(b = 6\), find the values of \(a\) for which the sequence is constant.
3. In the case where \(a = - 1\) and \(b = 8\), prove that \(U _ { n }\) is divisible by 7 for all even values of \(n\).

3
Given \(u _ { 1 } = 1\), determine which one of the formulae below defines an increasing sequence for \(n \geq 1\) Circle your answer.
[0pt] [1 mark]
\(u _ { n + 1 } = 1 + \frac { 1 } { u _ { n } } \quad u _ { n } = 2 - 0.9 ^ { n - 1 } \quad u _ { n + 1 } = - 1 + 0.5 u _ { n } \quad u _ { n } = 0.9 ^ { n - 1 }\)