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问题 解答题
已知各项均为正数的数列{an}的前n项和为Sn，且Sn，an，1成等差数列．
（1）求数列{an}的通项公式；
（2）若an2=2-bn，设Cn=
bn
an
求数列{Cn}的前项和Tn．
答案

（1）由题意2an=Sn+1，an＞0

当n=1时2a1=a1+1∴a1=1

n≥2时，sn=2an-1，sn-1=2an-1-1

两式相减an=2an-2an-1（n≥2）

整理得

an
an-1
=2（n≥2）（4分）

∴数列{an}1为首项，2为公比的等比数列．

∴an=a1•2n-1=1×2n-1=2n-1（5分）

（2）an2=2-bn=22n-2

∴bn=2-2n（6分）

Cn=

bn
an
=
2-2n
2n-1
=
4-4n
2n
Tn =
0
2
+
-4
22
+
-8
23
+…+
8-4n
2n-1
+
4-4n
2n
①

1
2
Tn=
0
22
+
-4
23
+…+
8-4n
2n
+
4-4n
2n+1
②

①-②

1
2
Tn=-4(
1
22
+
1
23
+…
1
2n
)  -
4-4n
2n+1
（9分）

=-4•

1
22
(1-
1
2n-1
)
1-
1
2
-
4-4n
2n+1
=-2(1-
1
2n-1
)-
4-4n
2n+1
=
n+1
2n-1
-2（11分）

∴Tn=

n+1
2n-2
-4（12分）

完形填空

Ⅲ．完形填空（每小题1.5分，共30分）

A seven-year-old boy , with his father , was inspecting a new car , one with a metal strip (条纹) looking like wood . “Dad , ”the boy    36     , “don’t buy this wood rot (烂东西).” He was shocked when his father started laughing . Then, when the father    37     his private remark with the salesman , and he laughed , the boy felt    38     .Some 30 years later , now a father himself , the man often thinks of this    39     whenever his own children say something that is unintentionally funny .

Most parents are     40    by such reactions because they don’t understand how deeply a child   41      shame . “All too often we don’t see or hear what is really    42     our children,” says Kaoru Yamamoto , a psychologist .

Yamamoto’s surveys    43    that children are afraid of losing    44     : being thought of as unattractive ,   45     or dishonest . It is more troubling to wet their pants in class , get a bad report card or    46     a grade than to be faced with a rival (竞争) baby brother or sister —situations a parent might     47    to be most disturbing . For a child , a blow to self-esteem , the sense of being worthy , is a     48    thing to stand .

Causing this fear , among other things , is the experience of being lost in a    49     place , of seeing their parents fight ,    50    when there is violence . About two-thirds of the children surveyed have experienced the    51     of parental fighting , which   52      them afraid that their mother and father are going to get divorced .

“In listening to children, it is   53      not to be touched by their strong sense of vulnerability (脆弱),” says Yamamoto. “They are afraid of being left alone , fearful of what lies   54      and uncertain of what they can do . They are completely dependent upon   55      in the face of upsetting and unpredictable events .”

36．A．shouted    B．whispered       C．cried        D．asked

37．A．told B．spared      C．shared      D．exchanged

38．A．pleased    B．afraid      C．surprised D．ashamed

39．A．occasion B．story        C．fun   D．event

40．A．delighted        B．puzzled    C．entertained      D．excited

41．A．faces       B．deals        C．fears        D．hates

42．A．interesting       B．troubling C．affecting D．hurting

43．A．show       B．inform     C．appear     D．prove

44．A．interest    B．courage    C．face D．freedom

45．A．unhealthy        B．energetic C．stupid      D．easy-going

46．A．repeat      B．fail   C．leave       D．change

47．A．wish        B．imagine   C．expect      D．realize

48．A．simple     B．rough      C．terrible    D．hopeless

49．A．strange    B．familiar   C．faraway   D．similar

50．A．mostly     B．naturally C．basically   D．especially

51．A．result       B．pain C．question   D．noise

52．A．leaves      B．lets   C．finds        D．gets

53．A．funny      B．true C．difficult   D．interesting

54．A．ahead      B．behind     C．above      D．around

55．A．teachers   B．neighbors        C．adults      D．friends

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单项选择题

The Turing machine is an abstract (71) of computer execution and storage introduced in 1936 by Alan Turing to give a mathematically precise definition of (72) . or ’mechanical procedure’. As such it is still widely used in theoretical computer science, especially in (73) theory and the theory of computation. The thesis that states that Turing machines indeed capture the informal notion of effective or mechanical method in logic and mathematics is known as Turing’s thesis.

Every Turing machine computes a certain (74) partial function over the strings over its alphabet. In that sense it behaves like a computer with a fixed program. However, as Alan luring already described, we can encode the action table of every Turing machine in a string. Thus we might try to construct a Turing machine that expects on its tape a string describing an action table followed by a string describing the input tape, and then computes the tape that the encoded Turing machine would have computed. As Turing showed, such a luring machine is indeed possible and since it is able to simulate any other Turing machine it is called a (75) Turing machine.

A universal Turing machine is Turing complete. It can calculate any recursive function, decide any recursive language, and accept any recursively enumerable language. According to the Church-Turing thesis, the problems solvable by a universal Turing machine are exactly those problems solvable by an algorithm or an effective method of computation, for any reasonable definition of those terms.

（72）处填（）。

A．operation

B．calculating

C．algorithm

D．mechanics
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