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)=
,数列{xn}的通项由xn=f(xn-1)(n≥2,n∈N+)确定.3x x+3
(Ⅰ)求证:{
}是等差数列; 1 xn
(Ⅱ)当x1=
时,求x100. 1 2
答
(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
∴{
}是等差数列;1 xn
(Ⅱ)由(Ⅰ)得
=2+(n-1)×1 xn
=1 3
n+5 3
∴x100=
=3 105
1 35