设f（x）=4^x/4^x+2(1）探究f(x)与f(1-x)的关系（2）求f(1/101)+f(2/1010）……f(100/101)

Question

设f（x）=4^x/4^x+2
(1）探究f(x)与f(1-x)的关系
（2）求f(1/101)+f(2/1010）……f(100/101)

Answer 1

f(x)=4^x/(4^x+2)=(4^x+2-2)/(4^x+2)=1-2/(4^x+2)
f(1-x)=1-2/(4^(1-x)+2)=1-2/[(4/4^x)+2]=1-2/[(4+2*4^x)/4^x]
=1-2*4^x/2(2+4^x)=1-4^x/(4^x+2)
所以 f(x)+f(1-x) =1-2/(4^x+2) +1-4^x/(4^x+2)=1
2 f(1/101)+f(2/101）……f(100/101)=f(1-100/101)+f(1-99/101)+.+f(99/100)+f(100/101)
=[f(1-100/101)+f(100/101)]+[f(1-2/101)+f(99/101)]+.+[f(1-50/101)+f(51/101)]
=50*1=50