高考若数列{an}满足,a1=1,且a(n+1)=an/(1+an),设数列{bn}的前n项和为Sn,且Sn=2-bn,求{bn/an}的前...高考若数列{an}满足,a1=1,且a(n+1)=an/(1+an),设数列{bn}的前n项和为Sn,且Sn=2-bn,求{bn/an}的前n项和Tn

Question

【高考】若数列{an}满足,a1=1,且a(n+1)=an/(1+an),设数列{bn}的前n项和为Sn,且Sn=2-bn,求{bn/an}的前...
【高考】若数列{an}满足,a1=1,且a(n+1)=an/(1+an),设数列{bn}的前n项和为Sn,且Sn=2-bn,求{bn/an}的前n项和Tn

Answer 1

1/a(n+1)=1/an+1
1/a1=1
1/an=n,
an=1/n
sn=2-bn,b1=1
sn=2-bn,s(n+1)=2-b(n+1)
b(n+1)=(1/2)*bn
bn=(1/2)^(n-1)
Tn=1+2*(1/2)^1+...+n*(1/2)^(n-1)
2Tn=2+2+...+n*(1/2)^(n-2)
Tn=4-(2+n)*(1/2)^(n-1)