The only way I seemed to be able to get it working is to use a function of
n+1 with the condition that n/2 elem Integers...
eg Simplify[(-1)^(n+1),n/2 elem integers]
then just have m=n+1, if you want to make it look neater..
eg Simplify[(-1)^(n+1)f[n+1],n/2 elem
I hope that is kinda what you wanted..