what you want to prove, is that if α is a n-cycle, with α = (a1 a2 ... an),
then αk sends aj to aj+k (mod n) (you can do this by induction on k).
now if n is even, say n = 2m, then α2 sends:
a1→a3→...→an-1→a1 (since n-1 = 2m -1 is odd)
that is α2 = (a1 a3 ... an-1)(a2 a4 ... an),
so α splits into 2 disjoint m-cycles.
but if n is odd, say n = 2m+1, so that n-2 is odd, then:
α2 = (a1 a3 .... an-2 an a2 a4 ... an-1),
which is an n-cycle.