Question 8: Let n 2 1 be an integer and consider a 1 x n board B, consisting of n cells, each one having side length one. The top part of the figure below shows By. 3 - - -. - - - R B W G We have an unlimited supply of bricks which are of the following types (see the bottom part of the figure above): . There are red ( R) and blue (B) bricks, both of which are 1 x 1 cells. . There are white (W) and green (G) bricks, both of which are 1 x 2 cells. A tiling of a board B, is a placement of bricks on the board such that: . the bricks exactly cover By, and . no two bricks overlap. In a tiling and colour can be used more than once and some colours might not be used at all. The figure below shows one possible tiling of By. W R B G B G Let T, be the number of different tilings of the board B,. (a) Determine 7, and T2- (b) Express Th recursively for n 2 3. (c) Prove that for any integer n 2 1, T _ (1+ v3)"- (1-\\/3)"+ 2V/3