问题 解答题
已知数列{an}的前项和Sn,当n≥2时,点(
1
Sn-1
1
Sn
)
在f(x)=x+2的图象上,且S1=
1
2

(1)数列{an}的通项公式;
(2)设bn=2(1-n)an求f(n)=
bn+2
(n+5)bn-1
的最大值及相应的n的值;
(3)在(2)的条件下当n≥2时,设Tn=
b22
+
b23
+…
b2n
.证明:Tn<1.
答案

(1)∵n≥2时,点(

1
Sn-1
1
Sn
)在f(x)=x+2的图象上,

1
Sn
-
1
Sn-1
=2,(n≥2)

故数列{

1
Sn
}是一个以2为公差的等差数列

又∵S1=

1
2
1
S1
=2

1
Sn
=2n,即Sn=
1
2n

∴n≥2时,an=Sn-Sn-1=

1
2n
-
1
2(n-1)
=
1
2(n-1)n

又∵n=1时,

1
2(n-1)n
无意义

故an=

1
2
,n=1
1
2(n-1)n
,n≥2

(2)∵bn=2(1-n)an

∴当n=1时,b1=0,

当n≥2时,bn=2(1-n)•

1
2(n-1)n
=
1
n

∴f(n)=

bn+2
(n+5)bn-1
=
n+1
(n+2)(n+5)
=
1
(n+1)+
4
n+1
+5
1
9

当且仅当n+1=2,即n=1时取等

(3)当n≥2时,

Tn=

b22
+
b23
+…
b2n

=

1
22
+
1
32
+…+
1
(n+1)2

1
1×2
+
1
2×3
+…+
1
n(n+1)

=1-

1
2
+
1
2
-
1
3
+…+
1
n
-
1
n+1

=1-

1
n+1
<1

即Tn<1

单项选择题
完形填空

     "Everything happens for the best," my mother said whenever I faced disappointment. "If you can   1  
on,
one day something good will happen. And you'll realize that it wouldn't have happened if not for that   2  
disappointment."
     Mother was right, as I discovered after graduating from college. I had   3   myself for a sports
announcer(播音员). So I went to Chicago and knocked on the door of every station and got   4   every
time.
     In one station, a kind lady told me that   40   stations wouldn’t risk employing a person without   5  
since I had just graduated. "Go out in the town and find a small station that might give you a chance,"she
said.
     I returned to Dixon,   6   I had finished my high school education and had   7   in the school football
team. My father said that our town had opened a store and wanted a local athlete to   8   its sports
department. The job sounded just   9   for me but I wasn’t hired.
     My   10   had shown as if by design. "Everything happens for the best," Mum   11   me. Dad offered
me his car to job hunt. I tried WOC Radio in Davenport. The program director told me they had already
hired a/an   12    13   I left his office, my frustration boiled over(发怒). I asked aloud, "  14   can a fellow
get to be a sports announcer if   15   can't get a job in a radio station?"
     Suddenly, I heard the director   16  . "What was that you said about sports? Do you know anything
about football?" Then he stood me before a microphone and asked me to   17   an imaginary game.  
     It was easy, and I did it without difficulty. The director was   18   and told me I would be broadcasting
a game on the   19   Saturday.

( )1. A. put            
( )2. A. later         
( )3. A. devoted        
( )4. A. invited        
( )5. A. large          
( )6. A. experience    
( )7. A. which         
( )8. A. played         
( )9. A. own           
( )10. A. possible      
( )11. A. disappointment
( )12. A. woke         
( )13. A. manager       
( )14. A. While        
( )15. A. What          
( )16. A. it            
( )17. A. calling       
( )18. A. make         
( )19. A. moved         
( )20. A. future        
B. take     
B. previous  
B. designed
B. promised  
B. all        
B. examination
B. where    
B. joined    
B. manage  
B. successful
B. delight  
B. advised  
B. clerk     
B. As        
B. How       
B. she      
B. answering
B. act       
B. sorry    
B. coming 
C. move      
C. frequent    
C. found      
C. refused  
C. good      
C. agreement  
C. there    
C. taken      
C. report    
C. right     
C. pleasure    
C. demanded    
C. worker    
C. Before    
C. Whenever  
C. he       
C. broadcasting
C. broadcast  
C. thankful    
C. last    
D. carry      
D. cautious    
D. enjoyed    
D. accepted    
D. small      
D. degree      
D. here        
D. fought      
D. enter      
D. important  
D. interest    
D. reminded    
D. announcer  
D. So          
D. Wherever    
D. I          
D. announcing  
D. play        
D. satisfied  
D. later