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我们知道对于形如{a_nb_n}的数列,其中{a_n}为等差数列,{b_n}为等比数列;这类数列通常采用“错位相减法”求和。由于等差数列通项为关于n的一次多项式f(n);如果f(n)为高于一次的多项式时,数列{a_nb_n}可拓广成为数列{f(n)b_n},其中f(n)为关于n的多项式b_n为等比数列。(公比q(?)1)这类的数列的和怎样求呢?如果仍用“错位相减法”求和运算相当繁琐。本文采用“待定系数法”及“构造法”的思想来研究此类问题,并给出一个简单求法,能收到事半功倍的效果。下面仅以几例说明供参考。
We know that for arrays of the form {a_nb_n}, where {a_n} is an arithmetic sequence and {b_n} is an analogous sequence; such sequences are usually summed using the “dislocation subtraction” method. Since the arithmetic progression general term is a primary polynomial f(n) for n; if f(n) is a polynomial higher than one, the sequence {a_nb_n} can be broadened into a sequence {f(n)b_n}, where f( n) The polynomial b_n for n is a geometric sequence. What is the sum of the series of numbers (compared to q(?)1)? If we still use the “dislocation subtraction method”, the sum operation is rather cumbersome. This paper adopts the ideas of “determined coefficient method” and “construction method” to study this type of problem, and gives a simple method to obtain a multiplier effect. Only a few examples are provided below for reference.