Some Relations of Generalized Fibonacci Sequence and Lucas Sequence

According to the definitions of generalized Fibonacci sequence and Lucas sequence, some relations of generalized Fibonacci and Lucas sequence were proved by using elementary method, such as, \sum\limits_i = 0^n u_iv_n - i = \left( n + 1 \right)u_n , 2^n + 1u_n + 1=\sum\limits_i = 0^n 2^iv_iA^n - i, \sum\limits_i = 0^n \left( - B \right)^iv_n - 2i = 2u_n + 1 , 3^n + 1u_n + 1 = \sum\limits_i = 0^n 3^iv_iA^n - i + \sum\limits_i = 0^n + 1 3^i - 1u_iA^n + 1 - i , \sum\limits_i = 0^n v_iv_n - i = \left( n + 1 \right)v_n + 2u_n + 1 = \left( n + 2 \right)v_n + Au_n, \left( A^2 + 4B \right)\sum\limits_i = 0^n u_iu_n - i = \left( n + 1 \right)v_n - 2u_n + 1 = nv_n - Au_n , and the relations of Fibonacci-Lucas sequences were promoted.