已知f(x)=3x/x+3数列{an}满足an=f(an-1),且a1=1/2,求an的通项公式

Question

已知f(x)=3x/x+3数列{an}满足an=f(an-1),且a1=1/2,
求an的通项公式

Answer 1

an = 3an-1/(an-1 + 3)
1/an = (an-1 + 3)/3an-1
=1/3 + 1/an-1
令bn = 1/an
则
bn = 1/3 + bn-1
故bn是公差为1/3的等差数列
a1 = 1/2
b1 = 2
故bn = (n+5)/3
an = 3/(n+5)