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arithmetic sequence wanted, (i - j) m_k + (j - k) m_i + (k - i) m_j = c ,

Source: 2020 Estonia TST 4.1

November 18, 2020
arithmetic sequencealgebraArithmetic Progression

Problem Statement

Let a1,a2,...a_1, a_2,... a sequence of real numbers. For each positive integer nn, we denote mn=a1+a2+...+annm_n =\frac{a_1 + a_2 +... + a_n}{n}. It is known that there exists a real number cc such that for any different positive integers i,j,ki, j, k: (i−j)mk+(j−k)mi+(k−i)mj=c(i - j) m_k + (j - k) m_i + (k - i) m_j = c. Prove that the sequence a1,a2,..a_1, a_2,.. is arithmetic
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