数列{an}满足a1=1,an+1=an+n+1(n∈N*),则1a1+1a2+…+1a2013等于()A.20122013B.40242013C.20131007D.10061007-数学

题目简介

数列{an}满足a1=1,an+1=an+n+1(n∈N*),则1a1+1a2+…+1a2013等于()A.20122013B.40242013C.20131007D.10061007-数学

题目详情

数列{an}满足a1=1,an+1=an+n+1(n∈N*),则
1
a1
+
1
a2
+…+
1
a2013
等于(  )
A.
2012
2013
B.
4024
2013
C.
2013
1007
D.
1006
1007
题型:单选题难度:中档来源:浙江二模

答案

由an+1=an+n+1得,an+1-an=n+1,
则a2-a1=1+1,
a3-a2=2+1,
a4-a3=3+1,

an-an-1=(n-1)+1,
以上等式相加,得an-a1=1+2+3+…+(n-1)+n-1,
把a1=1代入上式得,an=1+2+3+…+(n-1)+n=
n(1+n)
2

class="stub"1
an
=class="stub"2
n(n+1)
=2(class="stub"1
n
-class="stub"1
n+1
),
class="stub"1
a1
+class="stub"1
a2
+…+class="stub"1
a2013
=2[(1-class="stub"1
2
)+(class="stub"1
2
-class="stub"1
3
)+…+(class="stub"1
2013
-class="stub"1
2014
)]
=2(1-class="stub"1
2014
)=class="stub"2013
1007

故选C.

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