Question

已知a+b=1，a^2+b^2=2,求a^5+b^5的值

Answer 1

(a+b)²=1²
a²+b²+2ab=1
a²+b²=2
所以ab=-1/2

a³+b³=(a+b)(a²-ab+b²)
=(a+b)[(a+b)²-3ab]
=1*[1²+3/2)
=5/2

a^5+b^5
=(a²+b²)(a³+b³)-a²b³-a³b²
=(a²+b²)(a³+b³)-(ab)²(a+b)
=2*5/2-(1/4)*1
=19/4

Answer 2

a+b=1 平方
a^2+2ab+b^2=1
2ab=1-(a^2+b^2)=1-2=-1
ab=-1/2

a^2+b^2=2 平方：
a^4+2a^2b^2+b^4=4
a^4+b^4=4-2a^2b^2=4-2*(-1/2)^2=4-1/2=7/2

(a+b)(a^4+b^4)
=a^5+ab^4+a^4b+b^5
=a^5+b^5+ab(a^3+b^3)
a^5+b^5=(a+b)(a^4+b^4)-ab(a^3+b^3)
=(a+b)(a^4+b^4)-ab(a+b)(a^2-ab+b^2)
=1*7/2 - (-1/2)*1*(2-(-1/2))
=7/2 +1/2*1*5/2
=7/2+5/4
=19/4

Answer 3

因为a+b=1，a^2+b^2=2
所以1=(a+b)^2=a^2+b^2+2ab=2+2ab
所以ab=-1/2
所以a^3+b^3=(a+b)(a^2-ab+b^2)=1(2+1/2)=5/2
所以a^5+b^5=(a^3+b^3)(a^2+b^2)-a^2b^2(a+b)
=(5/2)2-1/4
=19/4

Answer 4

19/4