设正数列{an}的前n项之和是bn,数列{bn}的前n项之积是cn,若bn+cn=1

题目

设正数列{an}的前n项之和是bn,数列{bn}的前n项之积是cn,若bn+cn=1
(1)求a1,a2,a3,a4
(2)猜想数列{an}的通项公式
(3)求数列{an}的前n项和S

答案

正数列{an}的前n项之和是bnbn = a1 + a2 + …… + ancn = b1 * b2 * b3* …… bn令 n =1 b1 = a1c1 = b1因为 c1 + b1 = 1所以 a1 = b1 = c1 = 1/2b1 = a1b2 = a1 + a2c1 = b1 = a1c2 = b1*b2 = a1*(a1 + a2)b2 + c2 ...