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数学归納法是数学中一个很重要的工具,但初学者对它却是比较难于掌握的,如果学得不好,常常領会不到数学归納法的真意。一般,数学归納法可如下表述: 设有一个关于正整数n的命題,如果当n=1时該命題为真;又如果假設当n=k时該命題为真后,可以推出当n=k+1时該命題为真;那末該命題对一切正整数n为真。如果我們用“P(n)”表示一个含n的命題(n为正整数),那末上面一段話可以表述为: P(1) 真且由P(k)真可推得P(k+1)真,那末P(n)真。在上面那段用語言的敍述中,比較嚕嗦,难于一下子理解;在下面一段用式子的敘述中,比較清楚一些
Mathematical induction is a very important tool in mathematics, but beginners are more difficult to master it. If you do not learn well, you often cannot understand the true meaning of mathematical induction. In general, the mathematical induction can be expressed as follows: There is a proposition about the positive integer n, if the proposition is true when n=1, and if it is assumed that when n=k the proposition is true, it can be introduced when n= The proposition is true when k+1; then the proposition is true for all positive integers n. If we use “P(n)” to denote a proposition with n (n is a positive integer), then the above passage can be expressed as: P(1) True and P(k) can be derived as P(k+1) ) True, then P(n) is true. In the above narrative of the language, it is more difficult to understand at a glance; in the following description of the formula, it is clearer.