设数列{an}满足a1+2a2+3a3+······nan=n^2求数列{an}的通项公式

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设数列{an}满足a1+2a2+3a3+······nan=n^2求数列{an}的通项公式
设数列{an}满足a1+2a2+3a3+······nan=n^2求数列{an}的通项公式

设数列{an}满足a1+2a2+3a3+······nan=n^2求数列{an}的通项公式
a1+2a2+3a3+······nan=n^2
a1+2a2+3a3+······(n-1)an-1=(n-1)^2
两式相减 nan=n^2-(n-1)^2=2n-1
an=(2n-1)/n

a1+2a2+3a3+······+(n-1*an-1)+nan=n^2
a1+2a2+3a3+······+(n-1*an-1) =(n-1)^2
两者相减 得
n * an =n^2- (n-1)^2
an=2-1/n

当n≥2时,a1+2a2+3a3+…+nan=n²,a1+2a2+3a3+…+(n-1)an-1=(n-1)².
两式相减,得nan=2n-1,an=(2n-1)/n=2-1/n.
当n=1时,a1=1.
∴an=2-1/n.

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