Question

If T = $(5+2\sqrt{6}{)}^n$ = M + f , n \in N , 0 $\le$ f < 1 , Then M =

• A.
• B.
• C.
• D.

Answer 1

The correct option is C

$(5+2\sqrt{6}{)}^n$ = M + f , m $\in$ N , 0 $\le$ f < 1 .

Let $(5-2\sqrt{6}{)}^n$ = g , then 0 < g < 1 .

M + f + g = $(5+2\sqrt{6}{)}^n+(5-2\sqrt{6}{)}^n$

= 2 { \( {}^nC_05^n + {}^nC_2 5^{n-2} (2\sqrt{6})^2 + ... }

= integer

$\Rightarrow$ f + g is integer $\Rightarrow$

0 < f + g < 2 , f + g is integer $\Rightarrow$ f + g = 1

$\therefore$ (M+f)g = (5+2\sqrt{6})^n (5-2\sqrt{6})^n = 1 \)

$\Rightarrow$ M = $\frac{1}{g}-f=\frac{1}{1-f}-f$