问题 解答题
已知数列{an},a1=3,an+1=4an-3
(Ⅰ)设bn=1og2(an-1),求数列{bn}的前n项和Sn
(Ⅱ)在(Ⅰ)的条件下,求证:
1
S1
+
1
S2
+…+
1
Sn
n
n+1
答案

(Ⅰ)∵an+1=4an-3,∴an+1-1=4(an-1)

∵a1=3,∴a1-1=2,

∴{an-1}是以2为首项,4为公比的等比数列

∴an-1=2×4n-1=22n-1

∵bn=1og2(an-1),∴bn=2n-1,

∴数列{bn}是以1为首项,2为公差的等差数列

∴Sn=

n(1+2n-1)
2
=n2

(Ⅱ)证明:

1
S1
+
1
S2
+…+
1
Sn
=
1
12
+
1
22
+…+
1
n2
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
=
n
n+1

1
S1
+
1
S2
+…+
1
Sn
n
n+1

阅读理解

阅读理解。

The rich and the poor

  The world is divided (分为)into two main parts . The difference is that

one part is rich  and the other is poor . In the poor part , a lot of people never get

enough to eat . In the  rich part , a lot of people eat too much . In one part , child

starve (饿死)and in the other ,  a lot of people get fatter and fatter and have to

go on diets (吃特别饮食), or do special  exercises in order to lose weigt.     

  The poorer countries have special problems . Sometimes the land is too

poor to grow  anything on . The land can be improved (改善), but a lot of

things must be done first . The  people must be educated an water must be

found.     

  But rich countries have problems , too . There are not always pleasant

places to live in .  Sometimes the air is too dirty to breathe , and the rivers are

too dirty to swim in or to take  water from . The roads are too crowded to drive

along . Large numbers of people do not have  decent (象样的)housed to live

in . Some things will have to be done about these problems .  The air and the

river will have to be cleaned , and more houses will have to be built.     

Answer the questions :   

1. What’s the difference between the two parts ?     

___________________________  

2. Do the poorer countries have many problems ? What for example ?     

___________________________  

3. Can anything be done about it ?     

___________________________  

4. What must be done, for example ?     

___________________________  

5. Can all this be done easily ?     

___________________________

单项选择题