已知数列{an}中,a1=1,an+1=anan+1.(Ⅰ)求{an}的通项公式;(Ⅱ)若bn=2 1an-n,Sn=b1+b2+…+bn,求

2025-05-13 12:27:01
推荐回答(1个)
回答1:

(Ⅰ)∵an+1

an
an+1
,a1=1.
∴an≠0,
1
an+1
1
an
+1

1
an+1
?
1
am
=1

∴{
1
an
}是以1为首项,1为公差的等差数列.
1
an
1
a1
+(n?1)×1=n

an
1
n

(Ⅱ)bn2
1
an
?n
=2n-n,
∴Sn=b1+b2+…+bn
=(2+22+23+…+2n)-(1+2+…+n)
=2n+1?2?
n(n+1)
2

Sn?2n+1+47<0
2n+1?2?
n(n+1)
2
?2n+1+47<0

∴n2+n-90>0,
∴n>9或n<-10.
∵n∈N*
∴n>9.即n≥10.
∴n的最小值为10.