就 那條 發(fā)視頻的 一視頻 一命題 一般性 之證明
據(jù)
f(x+1)=-1/2f(x)+√3/2g(x)
g(x+1)=-1/2g(x)-√3/2f(x)
有
g(x+1)
=
-1/2(2√3/3f(x+1)+√3/3f(x)）
-√3/2f(x)
=
-√3/3f(x+1)-√3/6f(x)-√3/2f(x)
=
-√3/3f(x+1)-2√3/3f(x)
=
-√3/3(-1/2f(x)+√3/2g(x))
-2√3/3f(x)
=
√3/6f(x)-1/2g(x)-2√3/3f(x)
=
-√3/2f(x)-1/2g(x)
即
g(x+2)
=
-1/2g(x+1)-√3/2f(x+1)
=
-1/2(-√3/2f(x)-1/2g(x))
-√3/2(-1/2f(x)+√3/2g(x))
=
√3/4f(x)+1/4g(x)
+√3/4f(x)-3/4g(x)
=
√3/2f(x)-1/2g(x)
即
f(x+2)
=
-1/2f(x+1)+√3/2g(x+1)
=
-1/2(-1/2f(x)+√3/2g(x))
+√3/2(-1/2g(x)-√3/2f(x))
=
1/4f(x)-√3/4g(x)
-√3/4g(x)-3/4f(x)
=
-1/2f(x)-√3/2g(x)
即
f(x+3)
=
-1/2f(x+2)+√3/2g(x+2)
=
-1/2(-1/2f(x)-√3/2g(x))
+√3/2(√3/2f(x)-1/2g(x))
=
1/4f(x)+√3/4g(x)
+3/4f(x)-√3/4g(x)
=
f(x)
有
f(x+2)
=
-1/2f(x+1)+√3/2g(x+1)
=
1/4f(x)-√3/4g(x)
+√3/2(-1/2g(x)-√3/2f(x))
=
1/4f(x)-√3/4g(x)
-√3/4g(x)-3/4f(x)
=
-1/2f(x)-√3/2g(x)
且
g(x+2)
=
-1/2g(x+1)-√3/2f(x+1)
=
1/4g(x)+√3/4f(x)
-√3/2(-1/2f(x)+√3/2g(x))
=
1/4g(x)+√3/4f(x)
+√3/4f(x)-3/4g(x)
=
√3/2f(x)-1/2g(x)
即
g(x+3)
=
-1/2g(x+2)-√3/2f(x+2)
=
-√3/4f(x)+1/4g(x)
-√3/2(-1/2f(x)-√3/2g(x))
=
-√3/4f(x)+1/4g(x)
+√3/4f(x)+3/4g(x)
=
g(x)
即
g(2)=g(365)=-√3
據(jù)
f(x)=f(5-x)
設(shè)
x=2
有
f(2)=f(3)
且
f(3)=-1/2f(2)+√3/2g(2)
即
3/2f(2)=√3/2g(2)
即
√3f(2)=g(2)=-√3
即
f(2)=-1
即
f(3)=-1
有
g(x+1)
=
-√3/3f(x+1)-2√3/3f(x)
設(shè)
x=1
有
g(2)=-√3/3f(2)-2√3/3f(1)
即
-√3=√3/3-2√3/3f(1)
即
-4√3/3=-2√3/3f(1)
即
f(1)=2
即
Σ(k=1，2023)f(k)
=
674(f(1)+f(2)+f(3))+f(1)
=
674(2-1-1)+2
=
2
ps.
特殊
求解
除
視頻
擬定設(shè)法
抑或
設(shè)
f(x)=-2sin(2π/3x-7π/6)
g(x)=-2cos(2π/3x-7π/6)
求解過(guò)程如下
設(shè)
f(x)=Asin(ωx+φ)
g(x)=Acos(ωx+φ)
有
f(x+1)
=
Asin(ω(x+1)+φ)
=
Asin(ωx+ω+φ)
且
f(x+1)
=
-1/2f(x)+√3/2g(x)
=
cos(2π/3)Asin(ωx+φ)
+sin(2π/3)Acos(ωx+φ)
=
Asin(ωx+2π/3+φ)
即
ω=2π/3
據(jù)
f(x)=f(5-x)
設(shè)
x=5/2
2π/3x+φ=π/2
即
5π/3x+φ=π/2
即
φ=-7π/6
據(jù)
g(365)=-√3
有
Acos(730π/3-7π/6)
=
Acos(243π+π/3-7π/6)
=
Acos(243π-5π/6）
=
-Acos5π/6
=
√3A/2
=
-√3
即
A=-2
即
f(x)=-2sin(2π/3x-7π/6)
有
T=2π/(2π/3)=3
f(1)
=-2sin(2π/3-7π/6)
=2
f(2)
=-2sin(4π/3-7π/6)
=-1
f(3)
=-2sin(2π-7π/6)
=-1
即
Σ(k=1，2023)f(k)
=
674(f(1)+f(2)+f(3))+f(1)
=
674(2-1-1)+2
=
2
有關(guān)那條
是那什么
還想立牌坊
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