数列{an}的前n项和为Sn,a1=1,an+1-an-1=0,数列{bn}满足b1=2,anbn+1=2an+1bn.
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数列{an}的前n项和为Sn,a1=1,an+1-an-1=0,数列{bn}满足b1=2,anbn+1=2an+1bn.
(1)求S200; (2)求bn.
(1)求S200; (2)求bn.
(1)∵{an}的前n项和为Sn,a1=1,an+1-an-1=0,
∴an+1-an=1,
∴数列{an}是以a1=1为首项,d=1为公差的等差数列,
∴S200=200×1+
200×199
2×1=20100.
(2)由(1)得an=n,
∵数列{bn}满足b1=2,anbn+1=2an+1bn,
∴nbn+1=2(n+1)bn,
∴
bn+1
n+1=2•
bn
n,
∴{
bn
n}是以
b1
1=2为首项,q=2为公比的等比数列,
∴
bn
n=2×2n-1=2n,
∴bn=n•2n.
∴an+1-an=1,
∴数列{an}是以a1=1为首项,d=1为公差的等差数列,
∴S200=200×1+
200×199
2×1=20100.
(2)由(1)得an=n,
∵数列{bn}满足b1=2,anbn+1=2an+1bn,
∴nbn+1=2(n+1)bn,
∴
bn+1
n+1=2•
bn
n,
∴{
bn
n}是以
b1
1=2为首项,q=2为公比的等比数列,
∴
bn
n=2×2n-1=2n,
∴bn=n•2n.
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