问题 解答题
数列{an}各项均为正数,其前n项和为Sn,且满足2anSn-
a2n
=2

(Ⅰ)求证数列{
S2n
}
为等差数列,并求数列{an}的通项公式;
(Ⅱ)设bn=
2
4
S4n
-1
,求数列{bn}的前n项和Tn,并求使Tn
1
6
(m2-3m)
对所有的n∈N*都成立的最大正整数m的值.
答案

(Ⅰ)∵2anSn-

a2n
=1

当n≥2时,2(Sn-Sn-1)Sn-(Sn-Sn-1)2=1

整理得,

S2n
-
S2n-1
=1(n≥2),(2分)

S21
=1,(3分)

∴数列{

S2n
}为首项和公差都是1的等差数列.(4分)

S2n
=n,又Sn>0,∴Sn=
n
(5分)

∴n≥2时,an=Sn-Sn-1=

n
-
n-1

又a1=S1=1适合此式              (6分)

∴数列{an}的通项公式为an=

n
-
n-1
(7分)

(Ⅱ)∵bn=

2
4
S4n
-1
=
2
(2n-1)(2n+1)
=
1
2n-1
-
1
2n+1
(8分)

Tn=

1
1×3
+
1
3×5
+…+
1
(2n-1)(2n+1)

=1-

1
3
+
1
3
-
1
5
+…+
1
2n-1
-
1
2n+1

=1-

1
2n+1
=
2n
2n+1
(10分)

Tn

2
3
,依题意有
2
3
1
6
(m2-3m)
,解得-1<m<4,

故所求最大正整数m的值为3   (12分)

完形填空
完形填空。
                                                                   "See the doctor"
      In the waiting room,the patients (病人) , men and women,old and young,were sitting __1__ on the chairs, _2 _  for their turns. Billy, a school boy, was sitting there,too. They all looked very sad _ 3 _Billy. He _4_ an interesting story,and there was a smile on his face. Just then,the doctor came in to say he was ready for the next patient. Billy jumped up and ran into the __5 _  room.  
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实验题