proof that the pattern exists for all n Explain the pattern 
$$(\sqrt2-1)^1= \sqrt2 - \sqrt1$$
$$(\sqrt2-1)^2 = \sqrt9 - \sqrt8$$    
$$ (\sqrt2-1)^3   = \sqrt{50} - \sqrt{49} $$     
that is $(\sqrt2-1)^n$  is  equal to difference of two consecutive numbers one of which are squares.

This is related to the problem of finding which triangular numbers are a square, which leads to a Pell equation for $\sqrt 2$, and is related to the units of $\mathbb Z[\sqrt 2]$. Indeed, $(\sqrt2-1)^{-1}=\sqrt2+1$ is the fundamental unit.
The pattern is $(\sqrt2-1)^n = \sqrt{a_n+1} - \sqrt{a_n}$, where the $a_n$-th triangular number is a square. See A001108.
