I am really stuck with a few problems and would really appreciate any assistance possible. The problems I am having trouble with are:

1. Prove: For every integer n greater than or equal to 1, 1 + (1/2) + (1/4) + ... + (1/(2^(n-1))) = (2^(n) - 1) / (2^(n-1))

2. Prove: For every integer n greater than or equal to 5, 2^n > 2n +1

3. Prove: For every integer n greater than or equal to 0, 5^n - 3^n is an even integer

4. Prove: For every line of people with a woman at the front and a man at the rear, there is always an instance in which a woman immediately precedes a man.

Again, any help that anyone could provide on how to answer these problems would be incredibly helpful.