1. If (2-\sqrt{3})^n = a_n - b_n \sqrt{3} for all positive integers n, where a_n \ and \ b_n and integers, show that
(i) a_{n+1} = 2a_n + 3b_n \ and \ b_{n+1} = a_n + 2b_n
(ii) Calculate a_n^2 + 3b_n for n=1, 2 and 3.
(iii) Guess a formula for a_n^2-3b_n^2 and prove your guess is true for...