Question 127
Assume that $a$ is a positive integer and not a perfect square. Prove that for any positive integer $n$, the sum
$$
S_n = \{\sqrt{a}\} + \{\sqrt{a}\}^2 + \dots + \{\sqrt{a}\}^n
$$
is irrational, where $\{x\} = x - \lfloor x \rfloor$ and $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$.
$$
S_n = \{\sqrt{a}\} + \{\sqrt{a}\}^2 + \dots + \{\sqrt{a}\}^n
$$
is irrational, where $\{x\} = x - \lfloor x \rfloor$ and $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$.