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已知x²+y²-2x-4y+5=0,则1/xy+1/(x+1)(y+1)+1/(x+2)(y+2)+……+1/(x+2010)(y+2010)的值等于?
题目详情
已知x²+y²-2x-4y+5=0,则1/xy+1/(x+1)(y+1)+1/(x+2)(y+2)+……+1/(x+2010)(y+2010)的值等于?
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答案和解析
x²+y²-2x-4y+5=0
x²-2x+1+y²-4y+4=0
(x-1)²+(y-2)²=0
(x-1)²=0,(y-2)²=0
x=1
y=2
1/xy+1/(x+1)(y+1)+1/(x+2)(y+2)+……+1/(x+2010)(y+2010)
=1/1*2+1/2*3+1/3*4+.+1/2011*1/2012
=1-1/2+1/2-1/3+1/3-1/4+.+1/2011-1/2012
=1-1/2012
=2011/2012
x²-2x+1+y²-4y+4=0
(x-1)²+(y-2)²=0
(x-1)²=0,(y-2)²=0
x=1
y=2
1/xy+1/(x+1)(y+1)+1/(x+2)(y+2)+……+1/(x+2010)(y+2010)
=1/1*2+1/2*3+1/3*4+.+1/2011*1/2012
=1-1/2+1/2-1/3+1/3-1/4+.+1/2011-1/2012
=1-1/2012
=2011/2012
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