Question

Let $R=(8+3\sqrt{7}{)}^{20}$ and $\left[R\right]=$ the greatest integer less than or equal to $R$. Then which of the following is / are true

• A. $\left[R\right]$ is even
• B. $\left[R\right]$ is odd
• C. $R-\left[R\right]=1-\frac{1}{(8+3\sqrt{7}{)}^{20}}$
• D. None of these

Answer 1

Answer: (B), (C)

The correct options are
B $\left[R\right]$ is odd
C $R-\left[R\right]=1-\frac{1}{(8+3\sqrt{7}{)}^{20}}$

$R=\left[R\right]+g=(8+3\sqrt{7}{)}^{20}{=}^{20}{C}_0{8}^{20}{+}^{20}{C}_1{8}^{19}(3\sqrt{7})+......,where$
$0<g<1.$
$f=(8-3\sqrt{7}{)}^{20}{=}^{20}{C}_0{.8}^{20}{-}^{20}{C}_1{8}^{19}(3\sqrt{7})+......,where$
$0<f<1.$
$\therefore \left[R\right]+g+f=2{[}^{20}{C}_0{.8}^{20}{+}^{20}{C}_2{8}^{18}\left(3\sqrt{7}{)}^2\right]....]=$an even integer
$\therefore g+f$ is an integer $=1$

as $0<g+f<2$

since $0<g<1,0<f<1.$

$\therefore \left[R\right]=$ an even integer$-1=$an odd integer

Also, $R=\left[R\right]+g$

$R-\left[R\right]=g$

$R-\left[R\right]=1-f=1-(8-3\sqrt{7}{)}^{20}=1-{\frac{1}{(8+3\sqrt{7}{)}^{20}}}^{}$