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“问道于零,受福无量”,这是数学教育家傅种孙的名言.“0”的特殊地位和重要作用众所周知.高中数学新教材采用(W.Gellert)公理体系,将0归为自然数类.可见0与1、2、3…等自然数的和谐与统一.数列{an}的前n项和Sn,当n≥2时,Sn表示前n项的和,当n=1时,Sn表示a1,而S0是没有实际意义的.通过下面的研究,便可发现S0的妙用. 例1 已知数列{an}的前n项和为Sn=n2,则an=_. 解n=1时,a1=S1=1;n≥2时,an=Sn-Sn-1=n2-(n-1)2=2n-1. 又因为a1适合an=2n-1,所以an=2n-1(n ∈ N＊).
“Ask at zero, blessed and immeasurable.” This is the famous quote of math educator Fu Sun-son. The special status and important role of “0” is well known. The new textbook for high school mathematics adopts the (W.Gellert) axiom system, which classifies 0 as Natural numbers. See the harmony and unification of natural numbers such as 0 and 1, 2, 3, etc. The first n terms of the series {an} and Sn, when n≥2, Sn represents the sum of the first n terms, when n=1, Sn represents a1, and S0 is of no practical significance. Through the following research, we can discover the magical effect of S0. Example 1 Knowing that the first n terms of the series {an} are Sn=n2, an=_. Solution n= At 1 o’clock, a1=S1=1; n≥2, an=Sn-Sn-1=n2-(n-1)2=2n-1. Also because a1 is suitable for an=2n-1, an=2n- 1(n ∈ N*).