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.
| A | B | C | D |
| 1 | \(n\) | \(a _ { n }\) | \(b _ { n }\) | \(c _ { n }\) |
| 2 | 1 | 3 | 1.5 | 2.5 |
| 3 | 2 | -1 | 2.25 | 7.25 |
| 4 | 3 | 3 | 1.875 | 27.28125 |
| 5 | 4 | -1 | 2.0625 | 249.0889 |
| 6 | 5 | 3 | 1.96875 | 15512.32 |
| 7 | 6 | -1 | 2.01563 | 48126390 |
| 8 | 7 | 3 | 1.99219 | 3.86E+14 |
| 9 | 8 | -1 | 2.00391 | \(2.13 \mathrm { E } + 28\) |
| 10 | 9 | 3 | 1.99805 | 5.66E+55 |
| 11 | 10 | -1 | 2.00098 | 3.6E+110 |
Without attempting to solve any recurrence relations, describe the apparent behaviour, including as \(n \rightarrow \infty\), of
- \(a _ { n }\)
- \(\mathrm { b } _ { \mathrm { n } }\)
- \(\mathrm { C } _ { \mathrm { n } }\)