通过研究,得知 sum i=1 to n+1 a_ic_n~(i-1)的结果与数列有密切的关系,有以下二个定理:定理1:当数列{a_i}是等比数列时,sum i=1 to n+1 a_ic_n~(i-1)=a_i(1+q)~n证明如下:∵{a_i}是等比数列,不妨设公比为 qsum i=1 to n+1 a_ic_n~(i-1)=a_1c_n~0+a_2c_n~+1+a_3c_n~2+…+a_bc~(n-1)_n+a_(n+1)c~n_n=a_1c~0_n+a_1c~1_nq+a_1c~2_nq~2+…+a_1c~n_nq~n=a_1(1+q)~q Through the research, we know that the result of sum i = 1 to n + 1 a_ic_n ~ (i-1) is closely related to the sequence. There are two theorems as follows: Theorem 1: When the sequence {a_i} i = 1 to n + 1 a_ic_n ~ (i-1) = a_i (1 + q) ~ n is proved as follows: ∵ {a_i} is the geometric sequence, we can set the male ratio qsum i = 1 to n +1 a_ic_n ~ (n-1) = a_1c_n~0 + a_2c_n~ + 1 + a_3c_n~2 + ... + a_bc~n-1_n + a_n + 1 c~n_n = a_1c~0_n + a_1c~1_nq + a_1c ~ 2_nq ~ 2 + ... + a_1c ~ n_nq ~ n = a_1 (1 + q) ~ q