Question

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

• A.
• B.
• C.
• D.

Answer 1

The correct option is A

$LetS={(5-\sqrt{21})}^{2+1}R+S={(5+\sqrt{21})}^{2n+1}+{(5-\sqrt{21})}^{2n+1}=2{[}^{(2n+1)}{C}_{0}5{}^{2n+1}{+}^{(2n+1)}{C}_2{5}^{(2n-1)}21+---$= Even integer
$\Rightarrow \left[R\right]$+ f +S = Even integer
$0$< $5-\sqrt{21}$<$1\Rightarrow 0$<${(5-\sqrt{21})}^{2n+1}$< $1\Rightarrow 0$<$S$<$1$

f =$R-\left[R\right]\Rightarrow 0$<$f$< $1\Rightarrow 0f+S$ < $2\Rightarrow f+S=1\Rightarrow 1-f=S$

R $(1-f)$ = RS = ${(5-\sqrt{21})}^{2n+1}{(5-\sqrt{21})}^{2n+1}=4^{2n+1}$