因为 a(n+1)=(an +2)/an,
所以 a(n+1) +1 =2(an +1)/an
1/[a(n+1) +1] =an/[2(an +1)]
1/[a(n+1) +1] =(an +1 -1)/[2(an +1)]=1/2 -1/[2(an +1)]
令 bn=1/(an +1),则
b(n+1) =-(1/2)bn +1/2
b(n+1) -1/3 =-(1/2)(bn -1/3)
而b1 - 1/3=1/(a1+1) -1/3=1/6,
所以 {bn -1/3}是首项为1/6,公比为-1/2的等比数列,
所以 bn -1/3 =(1/6)(-1/2)^(n-1)=(-1/3)(-1/2)^n,
bn=(1/3)[1 -(-1/2)^n]
an=1/bn -1 =3/[1 -(-1/2)^n ] -1