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数列是高中数学的重要内容,近几年一直以把关题出现在高考数学江苏卷中,其核心知识“等差数列、等比数列”作为C级考点,显得尤为重要。在各类中学数学杂志中,对数列的前n项和S_n与通项a_n之间关系的研究比较多,而关于S_n~(1/2)与a_n之间关系的研究则较少。2015年镇江市高三数学期末考试出现了这样的试题:已知数列{a_n}中,a_1=1,在a_1,a_2之间插入1个数,在a_2,a_3之间插入2个数,在a_3,a_4之间插入3个数,…,在a_n,a_(n+1)之间插入n个数,使得所插入的数
The number sequence is an important part of high school mathematics. In recent years, the subject has been appearing in the Jiangsu mathematics college entrance examination, and its core knowledge of “equal difference series, equivalent series” is particularly important as a C-level test center. In all kinds of middle school mathematics magazines, there are many researches on the relationship between the first n terms and S_n and the general term a_n of the series, while there are few researches on the relationship between S_n ~ (1/2) and a_n. In 2015, Zhenjiang Senior High School mathematics final exam showed such questions as: a_1 = 1 is known in the series {a_n}, a number is inserted between a_1 and a_2, two numbers are inserted between a_2 and a_3, , a_4 is inserted between three numbers, ..., n numbers are inserted between a_n and a_ (n + 1) so that the number of inserted