let $(\begin{array}{l} a_{n + 1} \\ b_{n + 1} \end{array}) = (\begin{array}{cc} \sqrt{3} + 1 & 1 - \sqrt{3} \\ \sqrt{3} - 1 & \sqrt{3} + 1 \end{array}) (\begin{array}{l} a_{n} \\ b_{n} \end{array}), n = 1, 2, 3$ and a₁ = b₁ = 1 then a₂₀₁₀ =

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Detailed Solution

$\begin{array}{l} (\begin{matrix} a_{n + 1} \\ b_{n + 1} \end{matrix}) = 2 \sqrt{2} (\begin{array}{cc} \cos \frac{π}{12} & - \sin \frac{π}{12} \\ \sin \frac{π}{12} & \cos \frac{π}{12} \end{array}) (\begin{matrix} a_{n} \\ b_{n} \end{matrix}) \\ (\begin{matrix} a_{2010} \\ b_{2010} \end{matrix}) = (2 \sqrt{2})^{2009} {(\begin{array}{cc} \cos \frac{π}{12} & - \sin \frac{π}{12} \\ \sin \frac{π}{12} & \cos \frac{π}{12} \end{array})}^{2009} (\begin{matrix} 1 \\ 1 \end{matrix}) \\ (\begin{matrix} a_{2010} \\ b_{2010} \end{matrix}) = (2 \sqrt{2})^{2009} (\begin{array}{cc} \cos \frac{2009 π}{12} & - \sin \frac{2009 π}{12} \\ \sin \frac{2009 π}{12} & \cos \frac{2009 π}{12} \end{array}) (\begin{matrix} 1 \\ 1 \end{matrix}) \\ a_{2010} = (2 \sqrt{2})^{2009} (\cos (\frac{2009 π}{12}) - \sin (\frac{2009 π}{12})) \\ = (2 \sqrt{2})^{2009} \cdot \frac{1}{\sqrt{2}} = 2^{3013} \end{array}$