Interesting question

For too long, the problems on this website have been an endless bore of routine homework quesitons/calculations. Here is a question to provoke some thought:

For all $x$ such that $\{x\}+\frac1x=1$, is it necessarily true that $\{x^n\}+\frac1{x^n}=1$ for all nonnegative $n$? Here $\{x\}$ denotes the fractional part of $x$.