Consider the matrix

F =

0	1
1	1

Let i and j be any two integers. What is the product of the vector (i, j) and the matrix F? What happens if i and j are two consecutive numbers from the Fibonacci sequence? Use this idea to invent a divide-and-conquer algorithm to calculate this sequence, and analyse its efficiency, counting all arithmetic operations at unit cost. (We speak here of the time required to obtain only the n th element of the Fibonacci sequence, as a function of n, and not all the sequence)