19题,在数列{an}中,a1=2/3,3a n+1 -an-1=0.(1)求证:数列 {a n-1/2}为等比数列.(2)求数列{an}的通项

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19题,在数列{an}中,a1=2/3,3a n+1 -an-1=0.(1)求证:数列 {a n-1/2}为等比数列.(2)求数列{an}的通项
19题,在数列{an}中,a1=2/3,3a n+1 -an-1=0.(1)求证:数列 {a n-1/2}为等比数列.(2)求数列{an}的通项

19题,在数列{an}中,a1=2/3,3a n+1 -an-1=0.(1)求证:数列 {a n-1/2}为等比数列.(2)求数列{an}的通项
1
依题意:3a(n+1)-an-1=0,即:3a(n+1)-(an-1/2)-3/2=0
即:3(a(n+1)-1/2)=an-1/2,即:(a(n+1)-1/2)/(an-1/2)=1/3,且a1-1/2=1/6
所以:数列{an-1/2}是等比为1/3的等比数列
2
an-1/2=(a1-1/2)*(1/3)^(n-1)=(1/6)*(1/3)^(n-1)
所以,{an}的通项:an=(1/6)*(1/3)^(n-1)+1/2=(1/2)(1+1/3^n)