数列an满足a1=1/3,Sn=n(2n-1)an,求an

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Sn=n(2n-1)an,S(n+1)=(n+1)[2(n+1)-1]a(n+1)
=(n+1)[2n+1]a(n+1)
S(n+1)-Sn=a(n+1)
即(n+1)[2n+1]a(n+1)-n(2n-1)an=a(n+1)
n(2n-1)an=（2n^2+3n）a(n+1)
(2n-1)an=（2n+3）a(n+1)
a(n+1)=(2n-1)/（2n+3）an a2=1/5*1/3=1/15
=(2n-1)/（2n+3）*(2n-3)/（2n+1）a(n-1)
=(2n-1)/（2n+3）*(2n-3)/（2n+1）*(2n-5)/（2n-1）a(n-2)
.
=(2n-1)/（2n+3）*(2n-3)/（2n+1）*(2n-5)/（2n-1）*...*1/5a1
=(2n-1)/（2n+3）*(2n-3)/（2n+1）*(2n-5)/（2n-1）*...*1/5*1/3
=1/[（2n+3）（2n+1）]
an=1/[（2n+1）（2n-1）]

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