Proof of path dependence for savings with interest?

Hello!

Consider saving an amount X each year on an account with annual compunding. the interest rate is some interest rate process, or you can even consider a discrete set of values the interest can take every year. How do you prove that the final amount is path dependent? i.e. there's no amount X, for which the final amount is the same regardless of which path the interests take. Is this also true if the amount saved each year varies?

Thank for any help.