题目

已知函数f(x)=x/(3x+1),数列{an}满足a1=1,an+1=fan)(n∈N*),若数列{bn}的前n项和Sn=2^n-1,记Tn=b1/a1+b2/a2+.+bn/an,求Tn

提问时间：2020-11-26

答案

an+1 = an / (3an + 1)1/an+1 = 3 + 1/an令cn = 1/an则cn+1 = 3 + cnc1 = 1/a1 = 1故cn = 3n-2an = 1/(3n-2)Sn = 2^n - 1Sn-1 = 2^(n-1) - 1bn = Sn - Sn-1 = 2^n - 2^(n-1) = 2^(n-1)Tn = 1/1 + ...+ 2^(n-1)*(3n-2...