The given starting points for this inductive proof are the following:
1) the formulas, G(2x-2) = G(x)^2 - G(x-2)^2 and, G(2x-1) = G(x+1)G(x) - G(x-1)G(x-2)
or 2) the single formula for G(x) given above, in terms of the number a = ((5^.5) - 1)/2
So starting from 1), because 0 is the smallest natural, the base cases would be:
G(-2) = G(0)^2 - G(-2)^2 and G(-1) = G(1)G(0) - G(-1)G(-2)
But is this correct so far? I know my teacher defined natural numbers, in his class, to begin with x=0, but in this question he gave us, wouldn't it make more sense if the base cases started with x=1?
Also, G defines Fibonacci Numbers, and we are given the following:
G(0) = 0; G(1) = 1; for any x>=2, G(x) = G(x-1) + G(x-2)