已知数列{an}满足an+Sn=n,数列{bn}满足b1=a1,且bn=an-a(n-1),(n≥2）,试求数列{bn}

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已知数列{an}满足an+Sn=n,数列{bn}满足b1=a1,且bn=an-a(n-1),(n≥2）,试求数列{bn}的前n项的和Tn

an+Sn=n,a(n-1)+S(n-1)=n-1,前式减后式得：an-a(n-1)+an=1,2an-a(n-1)=1；2(an-1)=a(n-1)-1,(an-1)/[a(n-1)-1]=1/2,则数列{an-1}为等比数列公比q=1/2,因为a1+a1=1,首项为a1-1=-1/2,通项：an-1=-1/2*1/2^(n-1)=-1/2^n,则an=1-1/2^n；bn=an-a(n-1)=1-1/2^n-1+1/2^(n-1)=1/2^n；数列{bn}为首项=1/2,公比q=1/2的等比数列,前n项和Tn=1-1/2^n.

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