3 In this question you must show detailed reasoning.
- The sequence \(\left\{ A _ { n } \right\}\) is given by \(A _ { 1 } = \sqrt { 3 }\) and \(A _ { n + 1 } = ( \sqrt { 3 } + 1 ) A _ { n }\) for \(n \geqslant 1\).
Find an expression for \(A _ { n }\) in terms of \(n\).
- The sequence \(\left\{ B _ { n } \right\}\) is given by the formula
$$B _ { n } = \frac { 1 } { \sqrt { 3 } } \left( ( \sqrt { 3 } + 1 ) ^ { n } - ( \sqrt { 3 } - 1 ) ^ { n } \right) \text { for } n \geqslant 1 .$$
Explain why \(B _ { n } \rightarrow \frac { 1 } { \sqrt { 3 } } ( \sqrt { 3 } + 1 ) ^ { n }\) as \(n \rightarrow \infty\).
- The sequence \(\left\{ C _ { n } \right\}\) converges and is defined by \(C _ { n } = \frac { A _ { n } } { B _ { n } }\) for \(n \geqslant 1\).
Identify the limit of \(C n\) as \(n \rightarrow \infty\).