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.
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![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.](/uploads/image/z/1158740-44-0.jpg?t=%E5%B7%B2%E7%9F%A5%E6%95%B0%E5%88%97%7Ban%7D%E6%BB%A1%E8%B6%B3a1%3D4%2Can%3D4-4%2Fan-1%28n%3E%3D2%29%2C%E8%AE%BEbn%3D1%2Fan-2%281%29%E6%B1%82%E8%AF%81%7Bbn%7D%E6%98%AF%E7%AD%89%E5%B7%AE%E6%95%B0%E5%88%97%3B%282%29%E6%B1%82%E6%95%B0%E5%88%97%E7%9A%84%7Ban%7D%E7%9A%84%E9%80%9A%E9%A1%B9%E5%85%AC%E5%BC%8F.)
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