设函数f(x)=1/(4^x+2)⑴证明:对一切x∈R,f(x)+f(1-x)是常数;⑵an=f(0)+f(1/n)+f(2/n)+……+f((n－1）/n),求an,并求出数列{an}的前n项和.

Question

设函数f(x)=1/(4^x +2)
⑴证明:对一切x∈R,f(x)+f(1-x)是常数;
⑵an=f(0)+f(1/n)+f(2/n)+……+f((n－1）/n),求an,并求出数列{an}的前n项和.

Answer 1

f(x)+f(1-x)
=1/(4^x +2)+1/(4^(1-x) +2)
=1/(2^2x+2)+1/(2^(2-2x)+2)
=(2^(2-2x)+2+2^2x+2)/(2^2x+2)(2^(2-2x)+2)
=(2^(2-2x)+2^2x+2^2)/(2^2+2^(2-2x+1)+2^(2x+1)+2^2)
=(2^(2-2x)+2^2x+2^2)/(2^(2-2x+1)+2^(2x+1)+2^3)
=(2^(2-2x)+2^2x+2^2)/2(2^(2-2x)+2^(2x)+2^2)
=1/2
是常数
an=f(0)+f(1/n)+f(2/n)+……+f((n－1）/n)
=f(0)+f(1/n)+f(1-1/n)+f(2/n)+f(1-2/n)+...
=1/3+1/2*(n-1)/2
=1/3+(n-1)/4
=n/4+1/12
Sn=a1+a2+..+an=n/12+n(n+1)/8=n^2/8+5n/24