2 A binary operation * is defined on the set \(S = \{ p , q , r , s , t \}\) by the following composition table.
| \(*\) | \(p\) | \(q\) | \(r\) | \(s\) | \(t\) |
| \(p\) | \(p\) | \(q\) | \(r\) | \(s\) | \(t\) |
| \(q\) | \(q\) | \(p\) | \(s\) | \(t\) | \(r\) |
| \(r\) | \(r\) | \(t\) | \(p\) | \(q\) | \(s\) |
| \(s\) | \(s\) | \(r\) | \(t\) | \(p\) | \(q\) |
| \(t\) | \(t\) | \(s\) | \(q\) | \(r\) | \(p\) |
Determine whether ( \(S , *\) ) is a group.
- Find the general solution of
$$u _ { n } = 8 u _ { n - 1 } - 16 u _ { n - 2 } , n \geq 2 .$$
A new sequence \(v _ { n }\) is defined by \(v _ { n } = \frac { u _ { n } } { u _ { n - 1 } }\) for \(n \geq 1\).
- (A) Use \(( * )\) to show that \(v _ { n } = 8 - \frac { 16 } { v _ { n - 1 } }\). for \(n \geq 2\).
(B) Deduce that if \(v _ { n }\) tends to a limit then it must be 4 . - Use your general solution in part (i) to show that \(\lim _ { n \rightarrow \infty } v _ { n } = 4\).
- Deduce the value of \(\lim _ { n \rightarrow \infty } \left( \frac { u _ { n } } { u _ { n - 2 } } \right)\).