D
分析:本題就是將非零自然數(shù)n分別代入拋物線得出與x軸交點(diǎn)的各個值,分別算出兩交點(diǎn)間的距離再求出它們的和.
解答:將n=1,2,3,4…分別代入拋物線得y=x
2-
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x+
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,y=x
2-
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x+
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,y=x
2-
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x+
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,…;
分別解得x
1=1,x
2=
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,x
3=
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,x
4=
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,x
5=
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,x
6=
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,…;
∴A
1B
1=1-
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,A
2B
2=
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-
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,A
3B
3=
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-
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,…,A
2009B
2009=
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-
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;
∴A
1B
1+A
2B
2+…+A
2009B
2009=1-
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+
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-
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+
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-
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+…+
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-
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=1-
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=
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.
故選D.
點(diǎn)評:此題是一道開放題,需要先代入幾個特殊值,找出規(guī)律,然后解答.