The earliest evidence of using mathematical induction appeared in Francisco Morrico's Arithmetic Book (1575). Maurrico skillfully proved that the sum of the first n odd numbers is n 2 by using recursive relation, thus revealing the mystery of mathematical induction. The simplest and most common proof of mathematical induction is to prove that an expression holds when n belongs to all positive integers. This method consists of the following two steps: the basis of recursion: prove that the expression holds when n= 1 The basis of recursion: prove that it is true when n=m, and it is also true when n=m+ 1 The principle of this method is to first prove that the initial value in the expression is valid, and then prove that the proof process from one value to the next is effective. If these two steps are proved, then the proof of any value can be included in the repeated process. Maybe the domino effect is easier to understand. If you have a long row of vertical dominoes, then if you can be sure that the first domino will fall, as long as one domino falls, the next one will fall, then you can infer that all dominoes will fall. In this way, a recursive relationship is determined. As long as two conditions are met, all dominoes will fall: (1) The first domino will fall; (2) Any two adjacent dominoes will surely fall as long as the first one falls. In this way, no matter how many dominoes there are, as long as (1)(2) is guaranteed, they will all fall. Key points of solving problems: in the process of solving problems by mathematical induction, the conditions must be: hypothesis. Therefore, the hypothesis in inference can't be used as a known condition, and n=k+ 1 can't be directly substituted into the original formula of hypothesis in the following derivation process. It is necessary to substitute k+ 1 into the original known formula and further expand it to get the same expression as the original formula. Generally speaking, the structure is the same, and the original k can be completely replaced by k+ 1.