Question

设f(x)=4^x/[(4^x)+2],试求f(1/1001)+f(2/1001)+……f(1000/1001)

Answer 1

f(x)+f(1-x) =4^x/[(4^x)+2]+4^(1-x)/[4^(1-x)+2] =[4+2*4^X+4+2*4^(1-x)]/[4+2*4^X+4+2*4^(1-x)] =1 所以f(1/1001)+f(2/1001)+……f(1000/1001) =1/2{[f(1/1001)+f(1000/1001)]+[f(2/1001)+f(999/1001)]+···+[f(999/1001)+f(2/1001)]+[f(1000/1001)+f(1/1001) =1/2*1000 =500