Question

Let $R_n={(5\sqrt{5}+11)}^{2n+1}$ and $f_n=R_n-\left[R_n\right]$, where $\left[x\right]$ denotes the greatest integer $\le x$, find $\frac{1}{4}\left(R_3{f}_3\right)$.

Answer 1

$R_n\times f_n$
Implies $(5\sqrt{5}+11{)}^{2n+1}.(5\sqrt{5}-11{)}^{2n+1}$$=(125-121{)}^{2n+1}$$=4^{2n+1}$
For $n=3$,
$\Rightarrow R_3{f}_3=4^7$
Now
$\frac{1}{4}.\left(R_3{f}_3\right)$ $=\frac{4^7}{4}$$=4^6$$=2^{12}$
$=2^{10}\times 2$
$=1024\times 4$
$=4096$