作业帮数列 (24日 9:30:56)设数列{an}的前n项和sn=2an-1 [n=1,2,3…] 数列{bn}满足b1=3, bk+1=ak+bk [k=123…]求数列{bn}的前项和
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![数列 (24日 9:30:56)设数列{an}的前n项和sn=2an-1 [n=1,2,3…] 数列{bn}满足b1=3, bk+1=ak+bk [k=123…]求数列{bn}的前项和](/uploads/image/z/805656-48-6.jpg?t=%E6%95%B0%E5%88%97+%2824%E6%97%A5+9%3A30%3A56%29%E8%AE%BE%E6%95%B0%E5%88%97%7Ban%7D%E7%9A%84%E5%89%8Dn%E9%A1%B9%E5%92%8Csn%3D2an-1%26%23160%3B+%5Bn%3D1%2C2%2C3%E2%80%A6%5D%26%23160%3B+%E6%95%B0%E5%88%97%7Bbn%7D%E6%BB%A1%E8%B6%B3b1%3D3%2C%26%23160%3B+bk%2B1%3Dak%2Bbk%26%23160%3B%26%23160%3B+%5Bk%3D123%E2%80%A6%5D%E6%B1%82%E6%95%B0%E5%88%97%7Bbn%7D%E7%9A%84%E5%89%8D%E9%A1%B9%E5%92%8C)
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数列 (24日 9:30:56)设数列{an}的前n项和sn=2an-1 [n=1,2,3…] 数列{bn}满足b1=3, bk+1=ak+bk [k=123…]求数列{bn}的前项和
数列 (24日 9:30:56)
设数列{an}的前n项和sn=2an-1 [n=1,2,3…] 数列{bn}满足b1=3, bk+1=ak+bk [k=123…]求数列{bn}的前项和
数列 (24日 9:30:56)设数列{an}的前n项和sn=2an-1 [n=1,2,3…] 数列{bn}满足b1=3, bk+1=ak+bk [k=123…]求数列{bn}的前项和
sn=2an-1
s1=2a1-1=a1,a1=1
s(n-1)=2a(n-1)-1
an=2an-1-2a(n-1)+1
an=2a(n-1)
an=a1*2^(n-1)=2^(n-1)
b(k+1)=ak+bk=2^(k-1)+bk
b(k+1)-bk=2^(k-1)
bk-b(k-1)=2^(k-2)
b(k-1)-b(k-2)=2^(k-3)
……
b4-b3=2^2
b3-b2=2^1
b2-b1=2^0
两边相加：
bk-b1=2^(k-2)+2^(k-3)+2^(k-4)+……+2^2+2+1
=2^(k-1)-1
bk=2^(k-1)+2
设bn前n项和为Tn
Tn=[2^0+2]+[2^1+2]+[2^2+2]+……+[2^(n-3)+2]+[2^(n-2)+2]+[2^(n-1)+2]
=[2^0+2^1+2^2+……+2^(n-3)+2^(n-2)+2^(n-1)]+2n
=2^n-1+2n
Tn=2^n+2n-1