设数列{an},a1=3,a(n+1)=3an -2 (1)求证:数列{an-1}为等比数列

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设数列{an},a1=3,a(n+1)=3an -2 (1)求证:数列{an-1}为等比数列

设数列{an},a1=3,a(n+1)=3an -2 (1)求证:数列{an-1}为等比数列
设数列{an},a1=3,a(n+1)=3an -2 (1)求证:数列{an-1}为等比数列

设数列{an},a1=3,a(n+1)=3an -2 (1)求证:数列{an-1}为等比数列
方法一:A(n+1)-1=3An-3=3(An-1),且A1-1=2,所以数列{An-1}为公比为3,首项为2的等比数列
方法二:设A(n+1)+k=3(an+k),即A(n+1)=3An+2k,则2k=-2,k=-1,所以数列{An-1}为公比为3,首项为2的等比数列
方法三:由特征根方程得:x=3x-2,得x=1,所以{An-1}为等比数列且公比为3

递推式左右各-1,得到a(n-1)=3(an-1),且第一项不为0

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