KöMaL 2025 C1860

Let us consider a geometric progression consisting of positive integers. Prove that the contraharmonic mean of any three consecutive terms is an integer. How can the parity of this integer number be determined? The contraharmonic mean $C(a_{1},a_{2},\ldots,a_{n})$ of positive real numbers $a_{1}$, $a_{2}$, $\ldots$, $a_{n}$ is defined as $C(a_{1},a_{2},\ldots,a_{n})=\frac{a_{1}^{2}+a_{2}^{2}+\ldots+a_{n}^{2}}{a_{1}+a_{2}+\ldots+a_{n}}$.