Question

数列{An}满足a1=3/2,a (n+1 )=a(n)*2-a(n)+1,则m=1/a1+1/a2+1/a3+.+1/

Answer 1

a (n+1 )=a(n)*2-a(n)+1

得a(n+1)-1=an(an-1)

1/[a(n+1)-1]=1/(an-1)-1/an

得1/an=1/(an-1)-1/[a(n+1)-1]

所以m=1/a1+1/a2+1/a3+.+1/a2011

=1/(a1-1)-1/(a2-1)+1/(a2-1)-1/(a3-1)+……+1/(a2011-1)-1/(a2012-1)

=1/(a1-1)-1/(a2012-1)

=2-1/(2012-1)

由于从a3开始an就大于2,所以a2012-1>1故1/(a2012-1)