a+b=1,ab=(1/2)*{(a+b)^2-(a^2+b^2)=(1/2)*(1-2)= -1/2
(x-a)(x-b)=0; x有2根 x=a,x=b
(x-a)(x-b)=x^2-(a+b)x+ab=x^2-x-1/2=0
x^2=x+1/2
x^12=(x^2)^6=(x+1/2)^6=[(x+1/2)^3]*[(x+1/2)^3]=(x^6) + 6(x^5)((1/2)) + 15(x^4)((1/2)^2)+ 20(x^3)((1/2)^3) + 15(x^2)((1/2)^4) + 6(x^1)((1/2)^5) + ((1/2)^6)
x^11=(x^5) + 6(x^4)((1/2)) + 15(x^3)((1/2)^2) + 20(x^2)((1/2)^3)+ 15(x)((1/2)^4) + 6((1/2)^5) + ((1/2)^6)/x
x有2根 x=a,x=b
a^11=(a^5) + 6(a^4)((1/2)) + 15(a^3)((1/2)^2) + 20(a^2)((1/2)^3)+ 15(a)((1/2)^4) + 6((1/2)^5) + ((1/2)^6)/a----------------------------------------(1)
b^11=(b^5) + 6(b^4)((1/2)) + 15(b^3)((1/2)^2) + 20(b^2)((1/2)^3)+ 15(b)((1/2)^4) + 6((1/2)^5) + ((1/2)^6)/b---------------------------------------(2)
a+b=1,a^2+b^2=2 a^3+b^3=(a+b)(a^2+b^2)-ab(a+b)=5/2,a^4+b^4=(a^2+b^2)^2-2a^2b^2=7/2
a^5+b^5=(a+b)*(a^4+b^4)-ab(a^3+b^3)=19/4,(1/a)+(1/b)=(a+b)/ab=-2
从(1)+(2) ,和上面的等式
a^11+b^11=989/32