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问题 已知数列{an}的首项为a1=5,an= a1+a2+…+an-1(n≥2),求它的通项. 错解 由an=a1+a2+…+an-1=(a1+ a2+…+an-2)+an-1=an-1+an-1=2an-1得 an/an-1=2.故数列{an}是首项为a1=5,公比为2 的等比数列,所求的通项为an=5×2n-1. 分析 由已知a2=a1=5,但由an=5× 2n-1得a2=10,故为错解.出错的原因是对n的 范围注意不够,为了避免这种错误,在解题过 程中应注意以下两点:
The problem is that the first item of the series {an} is a1=5, an=a1+a2+...+an-1(n≥2), find its general term. The wrong solution consists of an=a1+a2+...+an-1 =(a1+ a2+...+an-2)+an-1=an-1+an-1=2an-1 get an/an-1=2. So the sequence {an} is the first item for a1=5, For a geometric sequence of 2, the general term sought is an=5×2n−1. The analysis is based on the known a2=a1=5, but a=5×2n-1 yields a2=10, which is the wrong solution. The reason for the error is insufficient attention to the range of n. In order to avoid this error, the following two points should be noted in the process of solving the problem: