Your question is ill defined. If you mean "What value of n satisfies this FOR ALL X?" then there isn't one. You can see this by working out the coefficient of [tex]x^{3}[/tex] in [tex](x+2)^{n}+(2-x)^{n}[/tex] and working out what value of n gives that coefficient as -1, so that it cancels with the [tex]x^{3}[/tex] term. Unfortunately, this leaves you with other terms which don't cancel (linear ones).
For any value of n you're going to find a value of x which satisfies the equation, though it will depend on n.
So basically no n gives it true for all x but any value of n makes it true for some x.
