设数列{an}满足a1+3a2+3^2a3+...+3^(n-1)an=n/3求an的通项

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设数列{an}满足a1+3a2+3^2a3+...+3^(n-1)an=n/3求an的通项
设数列{an}满足a1+3a2+3^2a3+...+3^(n-1)an=n/3求an的通项

设数列{an}满足a1+3a2+3^2a3+...+3^(n-1)an=n/3求an的通项
a1+3a2+3^2a3+...+3^(n-2)a(n-1)+3^(n-1)an=n/3 ①
a1+3a2+3^2a3+...+3^(n-2)a(n-1)=(n-1)/3 ②
①-②得
3^(n-1)an=(n/3)-[(n-1)/3]=1/3
因此an=(1/3)/[3^(n-1)]=1/(3^n)

a1+3a2+3^2a3+...+3^(n-1)an=n/3
把n换成n-1得
a1+3a2+3^2a3+...+3^(n-2)a(n-1)=(n-1)/3(n≥2)
两式相减得3^(n-1)an=1/3 an=1/3^n
又因a1=1/3也适合上式
∴an=1/3^n(n≥1)

a1+3a2+3^2a3+...+3^(n-1)an=n/3
a1+3a2+...+3^(n-2)an-1=n-1/3
an=(n/3-n-1/3)^n
=(1/3)^n

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