数列{an}的前n项和Sn=2an-1(n≥1）,数列{bn}满足b1=3,b(n+1)=an+bn,求数列{bn}的前n项的和Tn

题目

数列{an}的前n项和Sn=2an-1(n≥1）,数列{bn}满足b1=3,b(n+1)=an+bn,求数列{bn}的前n项的和Tn

答案

把a1带入sn得,a1=1,an=Sn-Sn-1=2an-2an-1,an/an-1=2;所以an=2^n-1;bn+1-bn=an=2^n-1,bn-bn-1=2^n-2……以此类推b2-b1=2^0把以上式子相加相消得,bn-b1=1+2^1+2^2+……+2^n-2bn=3+(1-2^n-1)/(1-2)=2^n-1 +2所以Tn=b1+b...