Question

The integral part of ${(5+2\sqrt{6})}^n$ where $n\in N$ is

• A. even
• B. odd
• C. irrational
• D. none of the above

Answer 1

Answer: (B)

odd

Let ${(5+2\sqrt{6})}^n=I+f$, where $I$ is an integer and $f$ is such that $0\le f<1$.Now, let $f^{\prime }={(5-2\sqrt{6})}^n,0<f^{\prime }<1$ Also $1+f+f^{\prime }={(5+2\sqrt{6})}^n+{(5-2\sqrt{6})}^n=2\{5^n+{{}^{n}C}_2{5}^{n-2}{\left(2\sqrt{6}\right)}^2+{{}^{n}C}_4{5}^{n-4}{\left(2\sqrt{6}\right)}^4+.....\}=2$ (Integer) $=2k(k\in N)$

Hence $f+f^{\prime }=2k-I$ is an integer, but $0<f+f^{\prime }<2$

$\therefore f+f^{\prime }=1$

$\therefore 1=2k-II=2k-1=$odd integer

Hence Integral part ${(5+2\sqrt{6})}^n$ is odd integer