A prime number $\displaystyle p$ is given. A sequence of positive integers $\displaystyle a_{1}, a_{2}, a_{3},...$ is determined by this conditon:

$\displaystyle a_{n+1}=a_{n}+p \lfloor \sqrt[p]{a_{n}} \rfloor$

Prove that there is a term in this sequence, which is a $\displaystyle p$-power of an integer number.