Well for the first part when M is an even number, when you put that -1 to those powers you will always get a positive 1, if you start with M=2 then is -1 times -1 which equals a positive 1 since negative time negative equals a positive. Therefore when you continue with higher even numbers the negative number will become positive.
For the second part, it is as you can imagine, the opposite, a negative number put to an odd power will stay negative. to explain this, imagine we go back to -1 to the power of 2 that equals a positive 1, but if you multiply it again by -1, so -1 to the power of 3 then that will give you a negative 1.
So in conclusion, when M= an even number, 2, 4, 6, -1 to those powers it will be positive
when M= an odd number, 1, 3, 5, -1 to those powers will stay negative.
