f(x)满足f(x+y)=f(x)×f(y),且f(1)=0.5.f(x)满足f(x+y)=f(x)×f(y),且f(1)=0.5_数学解答

f(x)满足f(x+y)=f(x)×f(y),且f(1)=0.5.
f(x)满足f(x+y)=f(x)×f(y),且f(1)=0.5.
1,当n∈N,求f(n)的表达式
2,设an=×f(n),n∈N,求证:a1+a2+a3+.+an＜2
3设bn=n×f(n+1)/f(n),n∈N,Sn为{bn}的前n项和,求1/S1+1/S2+...+1/Sn.
2,设an=n×f(n),n∈N,求证:a1+a2+a3+.....+an＜2

人气：207 ℃　时间：2020-05-08 12:46:40

解答

f(n)=f((n-1)+1)=f(n-1)*f(1)=f(n-1)*0.5=f(n-1)/2
同理 f(n+1)=f(n)/2
所以f(2)=f(1)/2f(n)=f(1)/2(n-1)(分母为2的n-1次方我打不出来)
f(n)=1/2n(分母为2的n次方我打不出来)
a1+a2+……an=a1+2a2+……nan=1/2+1/4+1/8+……+1/2n=(1/2+1/2n)*2/n=(n+1)/n=1+1/n
因为n∈N 所以1/n