Question 25
Let $a \gt 1$ be a non-integer number and $a \neq \sqrt[p]{q}$ for every positive integers $p \geq 2$ and $q \geq 1$, $k = [\log_{a} n] \geq 1$, where $[x]$ is the integral part of the real number $x$. Prove that for every positive integer $n \geq 1$ the equality
$$
[\log_{a} 2] + [\log_{a} 3] + \ldots + [\log_{a} n] + [a] + [a^{2}] + \ldots + [a^{k}] = n k
$$
holds.
$$
[\log_{a} 2] + [\log_{a} 3] + \ldots + [\log_{a} n] + [a] + [a^{2}] + \ldots + [a^{k}] = n k
$$
holds.