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在高中代数数学归纳法部份的教材中,曾举出反例a_n=(n~2-5n+5)~2来说明通过不完全归纳得出的结论要用数学归纳法证明的必要以及用数学归纳法证明命题的两个步骤缺一不可的道理。这个反例的作用是可想而知的。而具有探索精神的学生并不满足于仅仅知道这个反例,他们更想知道这个反例是怎样想出来的。为了培养学生思维的积极性和探索、创造的能力,我们进一步研究了这个反例的几种其他构造法。
In the textbook of algebraic mathematics induction in senior middle school, the counterexample a_n=(n~2-5n+5)~2 has been cited to illustrate the necessity of using mathematical induction to prove the conclusions obtained through incomplete induction, and to use mathematics. The inductive method proves that the two steps of the proposition are indispensable. The effect of this counterexample is conceivable. Students with an exploration spirit are not content to only know this counterexample. They are even more interested in knowing how the counterexample came up. In order to cultivate students’ enthusiasm for thinking and the ability to explore and create, we further studied several other construction methods of this counterexample.