Assume that x+(1/x) is an integer, how do I, by using induction, show that x^2 + (1/x^2) , x^3 + (1/x^3), .... , x^n + (1/x^n) are also integers?
Yes, it would make a difference : it would be false.
Let's call the relation " is an integer"
If one shows that if and are true then is true too, as and are true, we get that is true. Then, as and are true, we get that is true... so we'll reach any integer.
Instead, if you show that and true imply true, you go decreasing and won't reach all the integers...