定理　数列 {an}为等差数列的充要条件为 :对任意整数 k,当 m + 1≤ k≤ n - 1时 ,恒有等式 :( n - k) am + ( k - m) an =( n - m) ak,其中 m,n∈ N且 n m≥ 1 .证明　 (必要性 )设数列 {an}为等差数列 ,公差为 d,则　 an =am + ( n - m) d,于是对任意正整数 m,n,k有　 ( n - The necessary and sufficient condition for the theorem sequence {an} to be an arithmetic sequence is: for any integer k, when m + 1 ≤ k ≤ n - 1, the constant has an equality: (n - k) am + (k - m) an = (n - m) ak, where m, n ∈ N and nm ≥ 1. Prove (necessity) that the sequence {an} is an arithmetic sequence with a tolerance of d, then an = am + (n - m) d, So for any positive integer m,n,k there is (n -