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一般地,数列{a_n}若满足递归关系 a_n= ∫(a_(n-1),a_(n-2),…,a_(n-k)),那么它由递归关系及k个初始值确定,我们称其为递归数列。与递归数列有关的问题是数学竞赛中的一个热点。确定某些递归数列的通项在有关递归数列问题的研究中又占有重要地位,以下是几种常用方法。 1.代换法。例1 在数列{a_n}中,a_1=1,a_(n+1)=5a_n+1,求a_(n+1) 解依题设a_n+1=5a_n+1 ①以n代换n十1,可得 a_n=5a_(n-1)+1 ②①-②得a_(n+1)-a_n=5(a_n-a_(n-1))(n≥2) ③对③进行迭代,得
In general, if the sequence {a_n} satisfies the recursive relation a_n= ∫(a_(n−1), a_(n−2),..., a_(nk)), then it is determined by the recurrence relation and k initial values. Call it a recursive sequence. Problems associated with recursive sequences are a hot spot in the math competition. Determining the general terms of some recursive sequences also occupies an important position in the study of recursive sequence problems. The following are some common methods. 1. Substitution method. Example 1 In the sequence {a_n}, a_1=1,a_(n+1)=5a_n+1, find the a_(n+1) solution by setting a_n+1=5a_n+1 1 to replace n by 1 ,a_n=5a_(n-1)+1 21-2 is obtained a_(n+1)-a_n=5(a_n-a_(n-1))(n≥2) 3 Iterates over 3 to obtain 3