A.MATHS(Quadratic equations)

I use a = alpha, b =beta for easy typing

Prove by MI
1) n = 1 and n = 2
F(1) = (a-b) / 2^(3/2)
= [(a+b)^2 - 4ab] ^ 0.5 / 8^0.5
= (8/8) ^ 0.5
= 1 which is an integer
F(2) = (a^2 - b^2) / 8^0.5
= (a+b)(a-b) / 8^0.5
= (a+b)[(a+b)^2 - 4ab] ^ 0.5 / 8^0.5
= 2 * 1
= 2 which is an integer
so F(1) and F(2) are true

2) Assume n = k and n = k+1 are true
(a^k - b^k) / 2^1.5 = M
[a^(k+1) - b^(k+1)] / 2^1.5 = N
where M and N are integer

3) when n = k + 2
[a^(k+2) - b^(k+2)] / 8^0.5
= { [a^(k+1) - b^(k+1)](a+b) - ab(a^k - b^k) }/8^0.5
= [ N * 8^0.5 * (a+b) - ab * (M * 8^0.5) ] / 8^0.5
= [ (2N + M) * 8^0.5 ] / 8^0.5
= 2N + M
which is an integer
n = k+2 is true

By MI, n is true for all positive integers n