Riddles, brain puzzles and mathematical problems

[BAndrew]

Hint: Don't use mathematical induction. You can probably solve it, but it's way harder (+ there could be some flaws in your logic if you aren't careful because you aren't trying to prove something like (prove 1+2+3+..+n = n*(n+1)/2), but you want to prove that A is integer when n is odd). Try writing 1 + 1/n as (n+1)/n
The answer is 2 lines long (depends on how you write of course). With induction (as you saw) it already became complicated.