Question

a+b=1,ab=-1,a7次方+b7次方

Answer 1

a + b = 1，ab = - 1
S₁ = a + b = 1
S₂ = a² + b² = (a + b)² - 2ab = (1) - 2(- 1) = 3
S₃ = a³ + b³ = (a + b)(a² + b² - ab ) = (1)(3 - (- 1)) = 4
S₄ = a⁴ + b⁴ = (a²)² + (b²)² + 2a²b² - 2a²b² = (a² + b²)² - 2(ab)² = 3² - 2(- 1)² = 7

S(n) = aⁿ + bⁿ
S(n - 1) = aⁿ⁻¹ + bⁿ⁻¹
S(n - 2) = aⁿ⁻² + bⁿ⁻²

(a + b) · S(n - 1)
= (a + b) · (aⁿ⁻¹ + bⁿ⁻¹)
= aⁿ + bⁿ + ab(aⁿ⁻² + bⁿ⁻²)
= S(n) + ab · S(n - 2)
S(n - 1) = S(n) - S(n - 2) <==将a + b = 1，ab = - 1代入
S(n) = S(n - 1) + S(n - 2)
a⁷ + b⁷ = S(7)
= S(6) + S(5)
= [S(5) + S(4)] + [S(4) + S(3)]
= [S(4) + S(3) + S(4)] + [S(4) + S(3)]
= (7 + 4 + 7) + (7 + 4)
= 29

Answer 2

a+b=1
ab=-1
a²+b²=(a+b)²-2ab=1-2*(-1)=3

a^3+b^3=(a+b)^3-3ab(a+b)=1-3*(-1)*1=4
a^4+b^4=(a^2+b^2)^2-2(ab)^2=9-2*(-1)^2=7

a^7+b^7=(a^3+b^3)*(a^4+b^4)-(a^3b^4+a^4b^3)
=(a^3+b^3)*(a^4+b^4)-(ab)^3*(a+b)
=4*7-(-1)³*1
=29
呵呵！