已知函数f(x)=3x/x+3,数列{xn}的通项由xn=f(xn-1)(n≥2,n∈N+)确定. (Ⅰ)求证:{1/xn}是等差数列; (Ⅱ)当x1=1/2时,求x100.

问题描述:

已知函数f(x)=

3x
x+3
,数列{xn}的通项由xn=f(xn-1)(n≥2,n∈N+)确定.
(Ⅰ)求证:{
1
xn
}
是等差数列; 
(Ⅱ)当x1=
1
2
时,求x100

(Ⅰ)证明:∵xn=f(xn-1)=

3xn−1
xn−1+3

1
xn
=
1
3
+
1
xn−1

1
xn
1
xn−1
1
3

{
1
xn
}
是等差数列;
(Ⅱ)由(Ⅰ)得
1
xn
=2+(n-1)×
1
3
=
n+5
3

∴x100=
3
105
=
1
35