问题
解答题
.已知各项均为正数的数列{an}满足a1=1,an+1+an•an+1-an=0. (Ⅰ)求证:数列{
(Ⅱ)求数列{
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答案
(Ⅰ)∵an+1+an•an+1-an=0,
∴
=0,an+1+an•an+1-an an•an+1
∴
-1 an+1
=1,(3分)1 an
又
=1,1 a1
∴数列{
}是以1为首项,1为公差的等差数列.(4分)1 an
∴
=1+(n-1)×1=n,an=1 an
.(6分)1 n
(Ⅱ)由(Ⅰ)知
=n•2n.2n an
Sn=1×21+2×22+…+n×2n.①
2Sn=1×22+2×23+…+n×2n+1.②(9分)
由①-②得-Sn=21+22+…+2n-n×2n+1.
∴Sn=(n-1)2n+1+2.(12分)