I think it would be possible to prove it logically rather than through induction as well:
Prove floor(n/2) + floor(n/2 + 1/2) = n
LHS:
If n is even then n/2 = x where x is a whole number (integer)
Therefore floor(n/2) + floor(n/2 + 1/2)
= floor(x) + floor(x + 1/2)
= x + x
= 2x
= n
= RHS
If n is odd then n/2 = x + 1/2 where x is a whole number (integer)
Therefore floor(n/2) + floor(n/2 + 1/2)
= floor(x + 1/2) + floor(x + 1/2 + 1/2)
= x + x + 1
= 2x + 1
= 2(x + 1/2)
= n
= RHS
Therefore for any odd or even numbers it will be true (i.e. true for all integers). You can say that 0 is included under even numbers or prove it manually.
I think this is probably a better proof than induction considering that my proof for negative numbers by induction required this sort of logical analysis anyway. I guess it depends on the topic you're studying and whether they've specified a certain method.
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