Question

设函数f（x）＝2x-cosx,｛an｝是公差为8分之派的等差数列，f（a1）+f（a2）+……+f（a5）＝5派，则【f...

设函数f（x）＝2x-cosx,｛an｝是公差为8分之派的等差数列，f（a1）+f（a2）+……+f（a5）＝5派，则【f（a3）】的平方-a1×a3等于多少？

Answer 1

设a1=a
an=a+(n-1)*π/8
f(an)=2a+(n-1)*π/4-cos[a+(n-1)*π/8]
f(a1)+f(a2)+...+f(a5)=5(2a+2a+π)/2-cosa-cos(a+π/8)-cos(a+π/4)-cos(a+3π/8)-cos(a+π/2)
=10a+5π/2-[....]
=5π
因为π是无理数，所以a=π/4
a1=a=π/4
a3=a+2*π/8=π/2
[f(a3)]^2-a1*a3
=(π-0)^2-π/4*π/2
=(7π^2)/8

Answer 2

用图像解。可得f（a3）=f（2分之派），答案为八分之七派……