Think logically not mathematically [part-31]

\[a_{n} = \frac{1}{\sqrt{5}}\left[ \left(\frac{1+\sqrt{5}}{2}\right)^n-\left(\frac{1-\sqrt{5}}{2}\right)^n\right] \] Consider a sequence of real numbers \(\{a_n\}\) as given above, where \(n\) is a non-negative integer. Then which of the following are correct? \[\begin{array} {} \text{A)} & a_{5} & = & 5 \\ \text{B)} & a_{12} & = & 12^2 \\ \text{C)} & a_{10} & = & 55 \\ \text{D)} & a_{2009} + a_{2010} & = & 2 \ a_{2011} \end{array} \]

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