已知数列{an},满足a1=1/2,Sn=n²×an,求an

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已知数列{an},满足a1=1/2,Sn=n²×an,求an已知数列{an},满足a1=1/2,Sn=n²×an,求an已知数列{an},满足a1=1/2,Sn=n²×a

已知数列{an},满足a1=1/2,Sn=n²×an,求an
已知数列{an},满足a1=1/2,Sn=n²×an,求an

已知数列{an},满足a1=1/2,Sn=n²×an,求an

n≥2时,
Sn=n²×an S(n-1)=(n-1)²×a(n-1)
an=Sn-S(n-1)=n²×an-(n-1)²×a(n-1)
(n²-1)an-(n-1)²×a(n-1)=0
(n-1)[(n+1)an-(n-1)a(n-1)]=0
n≥2,n-1≥1,等式两边同除以n-1
(n+1)an-(n-1)a(n-1)=0
an/a(n-1)=(n-1)/(n+1)
a(n-1)/a(n-2)=(n-2)/n
…………
a2/a1=1/3
连乘
an/a1=(1/3)(2/4)...[(n-1)/(n+1)]=[1×2×...×(n-1)]/[3×4×...×(n+1)]=2/[n(n+1)]
an=2a1/[n(n+1)]=1/[n(n+1)]=1/n -1/(n+1)
n=1时,a1=1/1-1/2=1/2,同样满足通项公式
数列{an}的通项公式为an=1/n -1/(n+1).

可以求出 an=1/2n

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