Question

设f(x)=4^x/(4^x+2)求f(1/1001)+f(2/1001)+……+f(1000/1001)的值

Answer 1

f(x)+f(1-x)

=4^x/(4^x+2)+4^(1-x)/[4^(1-x)+2]

=4^x/(4^x+2)+(4/4^x)/[(4/4^x)+2]

=4^x/(4^x+2)+4/(4+2*4^x)

=4^x/(4^x+2)+2/(2+4^x)

=(4^x+2)/(4^x+2)

=1

所以f(1/1001)+f(2/1001)+……+f(1000/1001)

=[f(1/1001)+f(1000/1001)]+……+[f(500/1001)+f(501/1001)]

=1+1+……+1

=1*500

=500