Question 35
Consider a non-zero real number $a$ such that $\{a\} + \left\{ \frac{1}{a} \right\} = 1$, where $\{x\}$ denotes the fractional part of $x$. Prove that for any positive integer $n$, $\left\{ a^{n} \right\} + \left\{ \frac{1}{a^{n}} \right\} = 1$.