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数列求和问题是初等数学的重要内容之一,为充实传统的初等代数教材内容,本文仅就某些特殊数列的求和问题加以分类,探求前n项和的初等解法及理论根据。一、部分和变换法某些特定数列化为等差(或等比)数列求和十分方便,我们主要来看以下几种类型的问题。若{a_n}是等差数列,{b_n}是等比数列,那么怎样求数列{a_n±b_n}、{a_n b_n}及{a_n/b_n}或{b_n/a_n}的前n项的和呢? 我们可以利用变换部分和的方法来解,就是先将部分和进行“变换”,使数列转化为等差(或等比)数列的求和问题。例1 求下列数列的前n项的和:
The series summation problem is one of the important contents of elementary mathematics. In order to enrich the contents of the traditional elementary algebra teaching materials, this article only classifies the summation of certain special sequences, and seeks the elementary solution and the theoretical basis of the first n sums. Part I. Partial sum transformation method Some specific serializations are very convenient for sums of equal (or equal) numbers. We mainly look at the following types of problems. If {a_n} is an arithmetic sequence and {b_n} is a geometric sequence, then how do we find the sum of the first n terms of the sequence {a_n±b_n}, {a_n b_n}, and {a_n/b_n} or {b_n/a_n}? ? We can use the partial sum of the transformation to solve the problem, that is, the partial sum is “transformed” so that the sequence is transformed into a sum of arithmetic (or equal) series. Example 1 Find the sum of the first n terms of the following series: