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对偶思想是指,在求解数学问题时,根据题目中一个式子的结构特征,构造一个与之地位完全相伺,彼此间存在内在联系的对偶式,通过二者的协同作用,从而使问题获得巧妙解答.下面介绍几种常用方法,供参考.一、倒序对偶.把已知式的各部分施以倒序调节,所得式子称为已知式的倒序对偶式,再把它们对应部分相加(或相乘),促使问题解决.例1.证明:C_n~1+2C_n~2十3C_n~3十…+nC_n~n=n·2~(n-1)证明:设M=C_n~1+2C_n~2+3C_n~3+…+(n一1)C_n~(n-1)十nC_n~n,其倒序对偶式为:M’=nC_n~n+(n-1)C_n~n+(n-2)C_n~(n-2)+…+C_n~1两式相加得2M=nC_n~n+nC_n~(n-1)+nC_n~(n-2)+…+nC_n~1+nC_n~n=n(C_n~n+C_n~1+C_n~3+…+C_n~n)=n·2~n,∴M=n·2~(n-1).例2.求M=(1+tg1°)(1+tg2°)……(1+tg44°)的值解:注意到1°+44°=2°+43°=…=45°可构成M的倒序对偶式M’,M’=(1+tg44°)(1+tg43°)……(1+tg2°)(1+tg1°),两式相乘得:
Duality refers to solving a mathematical problem by constructing a duality based on the structural characteristics of a formula in the topic and constructing a duality that is in complete alignment with its intrinsic relationship. Through the synergy between the two, the problem is obtained. Ingenious solution. Here are some common methods, for reference. First, reverse order duality. The various parts of the known formula are adjusted in reverse order, the resulting formula is called the known reversed pair, and then add their corresponding parts (or multiplying) to cause the problem to be solved. Example 1. Proof: C_n~1+2C_n~2+3C_n~3+...+nC_n~n=n~2~(n-1) Proof: Let M=C_n~1 +2C_n~2+3C_n~3+...+(n−1)C_n~(n-1) ten nC_n~n, and its reversed pair is: M’=nC_n~n+(n-1)C_n~n+(n -2) C_n~(n-2)+...+C_n~1 Addition of 2M=nC_n~n+nC_n~(n-1)+nC_n~(n-2)+...+nC_n~1+nC_n ~n=n(C_n~n+C_n~1+C_n~3+...+C_n~n)=n·2~n,∴M=n·2~(n-1). Example 2. Find M=( 1+tg1°) (1+tg2°)... (1+tg44°) Value Solution: Note that 1°+44°=2°+43°=...=45° can constitute M’s reversed pair M’ M’=(1+tg44°)(1+tg43°)...(1+tg2°)(1+tg1°), the two formulas multiply: