Question 67
We are given a sequence $\langle a_1, a_2, a_3, \dots \rangle$ of real numbers. For every positive integer $n$ we define $m_n$ as the arithmetic mean of the numbers from $a_1$ through $a_n$. We assume that a real number $C$ exists, such that
$$
(i-j) \cdot m_k + (j-k) \cdot m_i + (k-i) \cdot m_j = C
$$
holds for all triples $(i, j, k)$ of pairwise different positive integers. Prove that $\langle a_1, a_2, a_3, \dots \rangle$ is an arithmetic sequence.
$$
(i-j) \cdot m_k + (j-k) \cdot m_i + (k-i) \cdot m_j = C
$$
holds for all triples $(i, j, k)$ of pairwise different positive integers. Prove that $\langle a_1, a_2, a_3, \dots \rangle$ is an arithmetic sequence.