Consider a sequence of real numbers as given above, where 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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a n = 5 1 [ ( 2 1 + 5 ) n − ( 2 1 − 5 ) n ] is the Binet's formula F n = 5 φ n − ψ n . Therefore, a n is the n th Fibonacci number F n . And a 5 = F 5 = 5 , a 1 2 = F 1 2 = 1 4 4 = 1 2 2 and a 1 0 = F 1 0 = 5 5 but a 2 0 0 9 + a 2 0 1 0 = F 2 0 0 9 + F 2 0 1 0 = F 2 0 1 1 = a 2 0 1 1 = 2 a 2 0 1 1 . Therefore, A, B, C are correct .