问题: 21. 当n为自然数时，有x^6n+1/x^6n=
21. 当n为自然数时，有x^6n+1/x^6n=2
证明：(1)x+1/x=-1 ==>x^6n+1/x^6n=2
(2)x+1/x=1 ==>x^6n+1/x^6n=2
22.证明：(1)a^2+b^2+c^2+d^2-ab-bc-cd-da=0 ==>a=b=c=d
(2)a^4+b^4+c^4+d^4-4abcd=0 ==>a=b=c=d
解答:
21. 当n为自然数时，有x^6n+1/x^6n=2
证明：(1)x+1/x=-1 ==>x^6n+1/x^6n=2
(2)x+1/x=1 ==>x^6n+1/x^6n=2
(1)x+1/x=-1----- x^2+1/x^2=-1
x^3+1/x^3=(x+1/x)(x^2+1/x^2-1)=(-1)(-1-1)=2
∴x^6n+1/x^6n=2
(2)x+1/x=1------ x^2+1/x^2=-1
x^3+1/x^3=(x+1/x)(x^2+1/x^2-1)=(-1)(-1-1)=2
∴x^6n+1/x^6n=2
22.证明：(1)a^2+b^2+c^2+d^2-ab-bc-cd-da=0 ==>a=b=c=d
(2)a^4+b^4+c^4+d^4-4abcd=0 ==>a=b=c=d
(1)a^2+b^2+c^2+d^2-ab-bc-cd-da=0
2(a^2+b^2+c^2+d^2-ab-bc-cd-da)=0
(a-b)^2+(b-c)^2+(c-d)^2+(d-a)^2=0
a-b=0,b-c=0,c-d=0,d-a=0.
∴a=b=c=d
(2)a^4+b^4+c^4+d^4-4abcd=0
(a^2-b^2)^2+(b^2-c^2)^2+(c^2-d^2)^2+(d^2-a^2)^2+2(a^2b^2+b^2c^2+c^2d^2+d^2a^2)-4abcd=0
(a^2-b^2)^2+(b^2-c^2)^2+(c^2-d^2)^2+(d^2-a^2)^2+(a^2b^2-c^2d^2)^2+(b^2c^2-d^2a^2)^2=0
a^2-b^2=0
b^2-c^2=0
c^2-d^2=0
d^2-a^2=0
a^2b^2-c^2d^2=0
b^2c^2-d^2a^2=0
∴a=b=c=d
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