Evaluation of lim Where n\in \mathbb{N}.
\bf{My\; Try::} Let \left(2+\sqrt{3}\right)^{2n} = I +f\;, where $0
Now Let 0<\left(2-\sqrt{3}\right)^{2n}<1\;, So \left(2-\sqrt{3}\right)^{2n}=f'.
So \left(2+\sqrt{3}\right)^{2n}+\left(2-\sqrt{3}\right)^{2n} = I +f+f' = \bf{Integer\; Quantity.}
So f+f'\in \mathbb{Z}. Now Given 0<(f+f')<2. So f+f' = 1\in \mathbb{Z}
So I+f+f' = \bf{Integer\; Quantity}\Rightarrow f = Integer\; Quantity-f'
So \displaystyle \lim_{n\rightarrow \infty}\left\{\left(2+\sqrt{3}\right)^{2n}\right\} = \bf{Integer\; quantity-}\displaystyle \lim_{n\rightarrow \infty}\left\{\left(2-\sqrt{3}\right)^{2n}\right\}
Now how can i solve after that, help me, Thanks
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