Question

数列{an}中,a1=1,a(n+1)=(an+2)/an

Answer 1

a(n+1)=(an+2)/an

a(n+1)-2 = (-an+2)/an

1/[a(n+1)-2] = -an/(an-2)

= -1 -2/(an-2)

1/[a(n+1)-2]+1/3 = -2[ 1/(an-2) +1/3]

[1/[a(n+1)-2]+1/3]/[ 1/(an-2) +1/3]=-2

[ 1/(an-2) +1/3]/[ 1/(a1-2) +1/3] = (-2)^(n-1)

1/(an-2) +1/3 = (1/3)(-2)^n

an -2 = 3/[ -1 +(-2)^n]

an = 2+3/[ -1 +(-2)^n]

bn = (an-2)/(an+1)

= {3/[ -1 +(-2)^n] }/{ 3+3/[ -1 +(-2)^n] }

= 1/ (-2)^n

=>{bn}是等比数列