a^5+b^5=（a+b)(a^4-a^3b+a^2b^2-ab^3+b^4)
∵a²=a+1,b²=b+1
∴a、b是方程x²-x-1=0的两根
a+b=1
ab=1
∴a^5+b^5
=（a+b)(a^4-a^3b+a^2b^2-ab^3+b^4)
=(a^4-a²+1-b²+b^4)
=(a+1)²-(a+1)+1-(b+1)+(b+1)²
=(a+1)(a+1-1)+1+(b+1)(b+1-1)
=a(a+1)+b(b+1)+1
=a²+a+b²+b+1
=a+1+a+b+1+b+1
=2(a+b)+1
=3