数列{an}中,a1=3,a(n+1)=3-(1/an-1)求证{1/(an-2)}是等差数列
问题描述:
数列{an}中,a1=3,a(n+1)=3-(1/an-1)求证{1/(an-2)}是等差数列
答
设{bn}:{1/(a(n)-2)};
即1/b(n)=a(n)-2;
1/b(n+1)=a(n+1)-2=3-(1/a(n)-1)-2=1-(1/a(n)-1)=(a(n)-2)/(a(n)-1)
b(n+1)-b(n)= (a(n)-1)/(a(n)-2) - 1/(a(n)-2) = 1
所以 证得