Question

Let $(1+\sqrt{2}{)}^n=x_n+y_n\sqrt{2}$ where $x_n,y_n$ are integers, then

• A. $x_n^2-2y_n^2=(-1{)}^n$
• B. $x_n+2y_n-x_{n+1}=3$
• C. $x_n^2-2y_n^2=1$
• D. $x_{n+1}-x_n-2y_n=1$

Answer 1

The correct option is A $x_n^2-2y_n^2=(-1{)}^n$
$(1+\sqrt{2}){}^n=x_n+\sqrt{2}{y}_n$ ...(i)
Let $(1-\sqrt{2}){}^n=x_n-\sqrt{2}{y}_n$ ...(ii)
Multiplying (i) with (ii), we get
$(1-2){}^n=x_n{}^2-2y_n{}^2$
$(-1){}^n=x_n{}^2-2y_n{}^2$
Hence, the answer is option A