已知ax+by=7,ax^2+by^2=49,ax^3+by^3=133,ax^4+by^4=406,求2008(x+y
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已知ax+by=7,ax^2+by^2=49,ax^3+by^3=133,ax^4+by^4=406,求2008(x+y)+2008xy+(a+b)/21的值
∵ax+by=7,
∴(ax+by)(x+y)=ax^2+axy+bxy+by^2=(ax^2+by^2)+xy(a+b).
又ax^2+by^2=49, ∴7(x+y)=49+xy(a+b).······①
∵ax^2+by^2=49,
∴(ax^2+by^2)(x+y)=ax^3+ax^2y+bxy^2+by^3=(ax^3+by^3)+xy(ax+by).
又ax^3+by^3=133,ax+by=7, ∴49(x+y)=133+7xy.······②
∵ax^3+by^3=133,
∴(ax^3+by^3)(x+y)=ax^4+ax^3y+bxy^3+by^4=(ax^4+by^4)+xy(ax^2+by^2).
又ax^4+by^4=406,ax^2+by^2=49, ∴133(x+y)=406+49xy.······③
联立:②、③,消去xy,得:
(133-49×7)(x+y)=406-133×7, ∴-210(x+y)=-525, ∴x+y=5/2.
将x+y=5/2代入②中,得:
49×(5/2)=133+7xy, ∴7xy=245/2-266/2=-21/2, ∴xy=-3/2.
将x+y=5/2、xy=-3/2代入①中,得:
7×(5/2)=49-(3/2)(a+b), ∴(3/2)(a+b)=98/2-35/2=63/2, ∴a+b=21.
于是:
2008(x+y)+2008xy+(a+b)/21
=2008[(x+y)+xy]+21/21
=2008×(5/2-3/2)+1
=2008+1
=2009
∴(ax+by)(x+y)=ax^2+axy+bxy+by^2=(ax^2+by^2)+xy(a+b).
又ax^2+by^2=49, ∴7(x+y)=49+xy(a+b).······①
∵ax^2+by^2=49,
∴(ax^2+by^2)(x+y)=ax^3+ax^2y+bxy^2+by^3=(ax^3+by^3)+xy(ax+by).
又ax^3+by^3=133,ax+by=7, ∴49(x+y)=133+7xy.······②
∵ax^3+by^3=133,
∴(ax^3+by^3)(x+y)=ax^4+ax^3y+bxy^3+by^4=(ax^4+by^4)+xy(ax^2+by^2).
又ax^4+by^4=406,ax^2+by^2=49, ∴133(x+y)=406+49xy.······③
联立:②、③,消去xy,得:
(133-49×7)(x+y)=406-133×7, ∴-210(x+y)=-525, ∴x+y=5/2.
将x+y=5/2代入②中,得:
49×(5/2)=133+7xy, ∴7xy=245/2-266/2=-21/2, ∴xy=-3/2.
将x+y=5/2、xy=-3/2代入①中,得:
7×(5/2)=49-(3/2)(a+b), ∴(3/2)(a+b)=98/2-35/2=63/2, ∴a+b=21.
于是:
2008(x+y)+2008xy+(a+b)/21
=2008[(x+y)+xy]+21/21
=2008×(5/2-3/2)+1
=2008+1
=2009
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