对于任意n∈N*,抛物线y=(n2+n)x2-(2n+1)x+1与x轴交于An,Bn两点,以|AnBn|表示该两点的距离,则|A1B1|+|A2B2|+…+|A1999B1999|的值是()A.199

题目简介

对于任意n∈N*,抛物线y=(n2+n)x2-(2n+1)x+1与x轴交于An,Bn两点,以|AnBn|表示该两点的距离,则|A1B1|+|A2B2|+…+|A1999B1999|的值是()A.199

题目详情

对于任意n∈N*,抛物线y=(n2+n)x2-(2n+1)x+1与x轴交于An,Bn两点,以|AnBn|表示该两点的距离,则|A1B1|+|A2B2|+…+|A1999B1999|的值是(  )
A.
1998
1999
B.
2000
1999
C.
1998
2000
D.
1999
2000
题型:单选题难度:中档来源:柳州三模

答案

y=(n2+n)x2-(2n+1)x+1=[x-class="stub"1
n
][x-class="stub"1
n+1
]
令y=0,则x=class="stub"1
n
class="stub"1
n+1

∴|AnBn|=class="stub"1
n
-class="stub"1
n+1

∴|A1B1|+|A2B2|+…+|A1999B1999|=(1-class="stub"1
2
)+(class="stub"1
2
-class="stub"1
3
)+…+(class="stub"1
1999
-class="stub"1
2000

=(1-class="stub"1
2
+class="stub"1
2
-class="stub"1
3
)+…+(class="stub"1
1999
-class="stub"1
2000
class="stub"1
2000

=1-class="stub"1
2000
=class="stub"1999
2000

故选D

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