已知数列{an}的各项均为正数,其前n项和为Sn,且Sn=an(an+1)2,n∈N*.(Ⅰ)求数列{an}的通项公式(Ⅱ)

2025-06-26 18:39:36
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回答1:

(Ⅰ)∵Sn

an(an+1)
2
,n∈N*,n=1时,a1S1
a1(a1+1)
2

∴a1=1或a1=0.又an>0,∴a1=1.
2Sn
a
+an,n∈N*
2Sn?1
a
+an?1,n≥2
,得2an=2(Sn?Sn?1)=
a
?
a
+an?an?1

∴(an+an-1)(an-an-1-1)=0,
∵an+an-1>0,∴an-an-1=1,n≥2,
∴数列{an}是以1为首项,1为公差的等差数列,
∴an=n…(6分)
(Ⅱ)bn=(2n?1)2n
Mn=1?2+3?22+5?23+…+(2n?1)?2n,…①
∴2Mn=1?22+3?23+…+(2n-3)?2n+(2n-1)?2n+1…②
由①-②得?Mn=1?2+2?22+23+…+2?2n?(2n?1)?2n+1
4?2n+2
1?2
?(2n?1)2n+1?2

Mn=(2n?3)?2n+1+6.…(12分)