若m^2=m+1,n^2=n+1,且m≠n,则m^5+n^5的值为

Question

若m^2=m+1,n^2=n+1,且m≠n,则m^5+n^5的值为

Answer 1

m^2=m+1,n^2=n+1.
m^2-n^2=(m+1)-(n-1)
(m+n)(m-n)=m-n
m≠n
m+n=1
(m+n)^2=m^2+n^2+2mn=1
m^2+n^2=(m+1)+(n+1)=m+n+2=1+2=3
2mn=1-(m^2+n^2)=1-3==-2
mn=-1
m^3+n^3=(m+n)(m^2+n^2-mn)=1*(3+1)=4
(m^2+n^2)(m^3+n^3)=m^5+m^2n^3+n^2m^3+n^3
3*4=m^5+n^5+m^2n^2(m+n)
m^5+n^5=12-(-1)^2*1=11