Let $R=(8+3√7)^{20}$ and [R]= the greatest integer less than or equal to R. Then A) [R] is even B) [R] is odd C) R-[R]=1-$\frac{1}{(8+3√7)^{20}}$ D) R+R[R]=1+$R^{2}$
It is a multiple choice question and i neef help as I am stuck.It has more than one choice
Let $a_n=(8+3\sqrt7)^{n}+(8-3\sqrt7)^{n}$. Then $$ a_0=2, \quad a_1=16 \quad a_{n}=16a_{n-1}-a_{n-2} \mbox{ for } n \ge 2 $$ In particular, $a_{n}$ is always an even integer.
Because $0 < 8-3\sqrt7 < 1$, the integer part of $(8+3\sqrt7)^{n}$ is $a_n-1$, which is always odd.
For the other options, note that $(8+3\sqrt7)(8-3\sqrt7)=1$ and more generally that they are the roots of $x^2=16x-1$ (hence the recurrence above).