Re: [中學] 請教算式解法

[Panthalassa (真。盤古大洋)]

※ 引述《sam206 ()》之銘言：
: 有幾題題目求算式
: 3)設實數滿足條件a+b=8
:                 ax+by=9
:                 ax^2+by^2=33
:                 ax^3+by^3=60
: 求ax^4+by^4

(a + b)(ax2 + by2)     = a2x2 + b2y2 + ab(x2 + y2)
(ax + by)(ax + by)     = a2x2 + b2y2 + ab(2xy)

=> ab(x2+y2-2xy)       = ab(x-y)^2
                       = 8*33-9*9 = 183  ... (1)

(a + b)(ax3 + by3)     = a2x3 + b2y3 + ab(x3 + y3)
(ax + by)(ax2 + by2)   = a2x3 + b2y3 + abxy(x + y)

=> ab(x3+y3-x2y-xy2)   = ab(x-y)^2(x+y)
                       = 8*60-9*33 = 183 ... (2)

(1) & (2) => x + y = 1


(a + b)(ax4 + by4)     = a2x4 + b2y4 + ab(x4 + y4)
(ax2 + by2)(ax2 + by2) = a2x4 + b2y4 + ab(2x2y2)

=> ab(x4 -2x2y2 +y4) = ab(x2 - y2)^2
                     = ab(x-y)^2 (x+y)^2
                     = 183
                     = 8*(ax4+by4) - 33*33

=> ax4 + by4 = (183 + 33*33)/8 = 159

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→ limil       : 這是一道經典一題多解題目。比較常見技巧是遞迴。