A Recurrnce Relation For Powers of Surds

We can rewrite powers of many surds as recurrence relations.
Take  
\[(2+3 \sqrt{5} )^n\]
  and write  
\[a_n+b_n \sqrt{5} = (2+3 \sqrt{5} )^n\]
.
Then
\[\begin{equation} \begin{aligned} a_{n+1}+b_{n+1} \sqrt{5} &= (2+3 \sqrt{5} )^{n+1} \\ &=(2+3 \sqrt{5} )(2+3 \sqrt{5} )^n \\ &=(2a_n+15b_n)+(3a_n+2b_n)\sqrt{5} \end{aligned} \end{equation}\]
.
We can then define  
\[(2+3   as
\[a_1=2, \: b_1=3\]

\[a_{n+1}=2a_n+15b_n, \: b_{n+1}=3a_n+2b_n\]
.

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