设数列{an}满足a1=13,an+1=an2+an(n∈N*),记Sn=11+a1+11+a2+…+11+an,则S10的整数部分为()A.1B.2C.3D.4-数学

题目简介

设数列{an}满足a1=13,an+1=an2+an(n∈N*),记Sn=11+a1+11+a2+…+11+an,则S10的整数部分为()A.1B.2C.3D.4-数学

题目详情

设数列{an}满足a1=
1
3
,an+1=an2+an(n∈N*),记Sn=
1
1+a1
+
1
1+a2
+…+
1
1+an
,则S10的整数部分为(  )
A.1B.2C.3D.4
题型:单选题难度:中档来源:不详

答案

∵数列{an}满足a1=class="stub"1
3
,an+1=an2+an=an(an+1)(n∈N*),
class="stub"1
an+1
=class="stub"1
an(an+1)
=class="stub"1
an
×class="stub"1
an+1
=class="stub"1
an
-class="stub"1
an+1

class="stub"1
an+1
=class="stub"1
an
-class="stub"1
an+1

S10=class="stub"1
a1
-class="stub"1
a2
+class="stub"1
a2
-class="stub"1
a3
+…+class="stub"1
a10
-class="stub"1
a11
=class="stub"1
a1
-class="stub"1
a11

a1=class="stub"1
3

a2=class="stub"1
9
+class="stub"1
3
 =class="stub"4
9

a3=class="stub"16
81
+class="stub"4
9
=class="stub"52
81

a4=class="stub"2704
6561
+class="stub"52
81
>1,
又an+1>an,
∴a11>1,
∴0<class="stub"1
a11
<1,
class="stub"1
a1
=3

∴S10的整数部分是2.
故选B.

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