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標題:

MQ61 --- Quartic Equations

發問:

• 我想要50個中外格言連解釋
• Maths!_1
• 會考英文要得幾多分先有level2-

• 我想要50個中外格言連解釋
• Maths!_1
• 會考英文要得幾多分先有level2-

 

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MQ61 --- Quartic EquationsDifficulty: 45%Givenax + by = 3,ax2 + by2 = 7,ax3 + by3 = 16,ax? + by? = 42.Evaluate ax? + by?.

最佳解答:

Let F(n) = ax^n + by^n Notice that (x + y)F(n) - xyF(n - 1) =(x + y)(ax^n + by^n) - xy(ax^n-1 + by^n-1) = ax^n+1 + bxy^n + ax^ny + by^n+1 - ax^ny - bxy^n = ax^n+1 + by^n+1 = F(n + 1) We have F(3) = (x + y)F(2) - xyF(1) => 16 = 7(x + y) - 3xy F(4) = (x + y)F(3) - xyF(2) => 42 = 16(x + y) - 7xy (x + y) = -14, xy = -38 So, ax^5 + by^5 = F(5) = (x + y)F(4) - xyF(3) = (-14)(42) - (-38)(16) = 20

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