已知x2=x+1,y2=y+1,且x≠y.(1)求证:x+y=1;(2)求x5+y5的值.
题型:解答题难度:一般来源:不详
已知x2=x+1,y2=y+1,且x≠y. (1)求证:x+y=1;(2)求x5+y5的值. |
答案
(1)证明:∵x2=x+1,y2=y+1, ∴x2-y2=x-y ∴x+y=1(x≠y) (2)∵x2=x+1,y2=y+1,∴x3=x2+x,y3=y2+y,x4=x3+x2,y4=y3+y2,x5=x4+x3,y5=y4+y3, ∴x5+y5 =x4+x3+y4+y3 =x3+x2+x2+x+y3+y2+y2+y, =x2+x+x2+x2+x+y2+y+y2+y2+y, =3(x2+y2)+2(x+y), =3(x+1+y+1)+2(x+y), =3×3+2, =11. |
举一反三
若m2+m-1=0,则m3+2m2+2004=______. |
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