a1=2,an=2-1/an-1(n≥2),令bn=1/an-1已知数列{an}满足a1=2,an=2-1/a(n-1)(n≥2),令bn=1/(an-1)求证：数列{bn}是等差数列

Question

a1=2,an=2-1/an-1(n≥2),令bn=1/an-1
已知数列{an}满足a1=2,an=2-1/a(n-1)(n≥2),令bn=1/(an-1)
求证：数列{bn}是等差数列

Answer 1

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/(2-1)=1
数列{1/(an -1)}是以1为首项,1为公差的等差数列.
又bn=1/(an -1),数列{bn}是以1为首项,1为公差的等差数列.
b1=1/(a1 -1)=1
bn=1+(n-1)=n
数列{bn}的通项公式为bn=n.