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“数列”是中学数学的重要内容之一,而“数列求和,”形式多样,变化无穷。中学生大都对明显的等差数列,等比数列求和掌握熟练,应用自如;但对其它数列的求和问题,往往感到无能为力。怎样才能解决这个矛盾呢?根据笔者的教学实践,摸索出求特殊类型数列之和的五种方法,现举例说明如下。一、还差法例1 求sum from k=1 to n (1/k(k+1)(k+2)) 不难发现:数列各项的分母,是等差数列连续三项之积:并且前两项之积与后两项之积的倒数的公差是2,其通项可变形为
“Number series” is one of the important contents of middle school mathematics, and “the summation of numbers” has various forms and infinite changes. Most of the middle school students are indifferent to the obvious arithmetic series, the mastery of the mathematical series and the ease of application, but they often feel powerless about the summation of other series. How can we resolve this contradiction? According to the author’s teaching practice, the five methods for finding the sum of special types of numbers are explored. The examples are as follows. First, the difference law 1 seeks sum from k = 1 to n (1/k(k+1)(k+2)). It is not difficult to find that the denominator of each sequence is the product of three consecutive terms of the arithmetic sequence: The tolerance between the product of the first two items and the inverse of the product of the last two items is 2, and its general term can be transformed into