已知数列[an]满足An+1=1+an /3-an ,且a1=1/3,求证数列[1/(an -1)]是等差数列,并求an
问题描述:
已知数列[an]满足An+1=1+an /3-an ,且a1=1/3,求证数列[1/(an -1)]是等差数列,并求an
答
a(n+1)=1+an/3-an
a(n+1)-1=1+an/3-an-1
=1+an-3+an/3-an
=2an-2/3-an
则1/[a(n+1)-1]=3-an/2an-2
=2/2an-2+ 1-an/2an-2
=1/(an-1) - 1/2
即1/[a(n+1)-1]-1/(an-1)= - 1/2
为等差数列
算出1/(an-1),再反求an就ok拉