m^2=m+1,n^2=n+1,且m/=n.求m^5+n^5的值

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m^2=m+1,n^2=n+1,且m/=n.求m^5+n^5的值
m^2=m+1,n^2=n+1,且m/=n.求m^5+n^5的值

m^2=m+1,n^2=n+1,且m/=n.求m^5+n^5的值
m^2=m+1,n^2=n+1令方程x²=x+1,则m,n是它的两根,所以m+n=1
m^2=m+1,m^4=m^2+2m+1=3m+2同理n^4=3n+2
m^5+n^5
=m(3m+2)+n(3n+2)
=3m^2+2m+3n^2+2n
=5m+2+5n+2
=5(m+n)+4
=5+4
=9