Question

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.（

Answer 1

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)