已知数列{an}中,a1=3/5,an=2-1/an-1(n》2),数列{bn)满足bn=1/an-1.求证数列{bn}是等差数列.

问题描述:

已知数列{an}中,a1=3/5,an=2-1/an-1(n》2),数列{bn)满足bn=1/an-1.求证数列{bn}是等差数列.

证:
an=2-1/a(n-1)
an - 1= 1 - [1/a(n-1)]=[a(n-1)-1]/a(n-1)
1/(an - 1)=a(n-1)/[a(n-1)-1]=[a(n-1)-1+1]/[a(n-1)-1]=1+1/[a(n-1)-1]
1/(an - 1)-1/[a(n-1)-1]=1,为定值.
1/(a1-1)=1/(3/5-1)=-5/2
bn=1/(an-1)
数列{bn}是以-5/2为首项,1为公差的等差数列