If  $\text{R=(5+}\sqrt{\text{21}}{\text{)}}^{\text{2n+1}}$and$\text{f=R-[R]}$ , where [R] denotes the greatest integer less than or equal to R, then R(1-f)=

$\begin{array}{l}\text{ Let }\mathrm{S}=(5+\sqrt{21}{)}^{2n+1}\\ \mathrm{R}+\mathrm{S}=(5+\sqrt{21}{)}^{2n+1}+(5+\sqrt{21}{)}^{2n+1}=2\left[{}^{(2n+1)}\mathrm{C}_0{5}^{2\mathrm{n}+1}+{}^{(2n+1)}\mathrm{C}_1{5}^{2n-1}\left(21\right)\right]\end{array}$

=Even integer$\Rightarrow \left[\text{R}\right]+\text{f+S}=$Even Integer

$\begin{array}{l}0<5-\sqrt{21}<1\Rightarrow 0<(5-\sqrt{21}{)}^{2n+1}<1\Rightarrow 0<\mathrm{S}<1\\ f=R-\left[\mathrm{R}\right]\Rightarrow 0<\mathrm{f}<1\Rightarrow 0<\mathrm{f}+\mathrm{S}<2\Rightarrow \mathrm{f}+\mathrm{S}=1\Rightarrow 1-\mathrm{f}=\mathrm{S}\\ \mathrm{R}(1-\mathrm{f})=\mathrm{RS}=(5+\sqrt{21}{)}^{2n+1}(5-\sqrt{21}{)}^{2n+1}=4^{2\mathrm{n}+1}\end{array}$