已知数列{an}满足a1=4,an=4-4/an-1(n>=2),设bn=1/an-2(1)求证{bn}是等差数列;(2)求数列的{an}的通项公式.

已知数列{an}满足a1=4,an=4-4/an-1(n>=2),设bn=1/an-2(1)求证{bn}是等差数列;(2)求数列的{an}的通项公式.
已知数列{an}满足a1=4,an=4-4/an-1(n>=2),设bn=1/an-2(1)求证{bn}是等差数列;(2)求数列的{an}的通项公式.

已知数列{an}满足a1=4,an=4-4/an-1(n>=2),设bn=1/an-2(1)求证{bn}是等差数列;(2)求数列的{an}的通项公式.
(1)证明：an-2=2-4/a(n-1)=(2a(n-1)-4)/a(n-1)
1/(an-2)=a(n-1)/(2a(n-1)-4)=1/2*a(n-1)/(a(n-1)-2)=1/2[1+2/(a(n-1)-2)]
所以bn=1/2(1+2b(n-1))=b(n-1)+1/2
即{bn}为等差数列,首项1/(a1-2)=1/2,公差为1/2
(2)bn=n/2
即1/(an-2)=n/2
所以an=2/n+2