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问题 解答题
已知{an}是首项为2，公比为
1
2
的等比数列，Sn为它的前n项和．
（1）用Sn表示Sn+1；
（2）是否存在自然数c和k，使得
Sk+1-c
Sk-c
＞2成立．
答案

解（1）由Sn=4(1-

1
2n
)，得Sn+1=4(1-
1
2n+1
)=
1
2
Sn+2(n∈N)．

（2）要使

Sk+1-c
Sk-c
＞2，只要
c-(
3
2
Sk-2)
c-Sk
＜0．

因为Sk=4(1-

1
2k
)＜4，所以Sk-(
3
2
Sk-2)=2-
1
2
Sk＞0(k∈N)，

故只要

3
2
Sk-2＜c＜Sk(k∈N)．①

因为Sk+1＞Sk（k∈N），所以

3
2
Sk-2≥
3
2
S1-2=1，

又Sk＜4，故要使①成立，c只能取2或3．

当c=2时，因为S1=2，所以当k=1时，c＜Sk不成立，从而①不成立．

因为

3
2
S2-2=
5
2
＞c，由Sk＜Sk+1（k∈N），得
3
2
Sk-2＜
3
2
Sk+1-2，所以当k≥2时，
3
2
Sk-2＞c，从而①不成立．

当c=3时，因为S1=2，S2=3，

所以当k=1，2时，c＜Sk不成立，从而①不成立．

因为

3
2
S3-2=
13
4
＞c，又
3
2
Sk-2＜
3
2
Sk+1-2，

所以当k≥3时，

3
2
Sk-2＞c，从而①不成立．

故不存在自然数c、k，使

Sk+1-c
Sk-c
＞2成立．

单项选择题

In almost all cases the soft parts of fossils are gone for ever but they were fitted around or within the hard parts. Many of them also were attached to the hard parts and usually such attachments are visible as depressed or elevated areas, ridges, or grooves, smooth or rough patches on the hard parts. The muscles most important for the activities of the animal and most evident in the appearance of the living animal are those attached to the hard parts and possible to reconstruct from their attachments. Much can be learned about a vanished brain from the inside of the skull in which it was lodged.
Restoration of the external appearance of an extinct animal has little or no scientific value. It does not even help in inferring what the activities of the living animal were, how fast it could run, what its food was, or such other conclusions as are important for the history of life. However, what most people want to know about extinct animals is what they looked like when they were alive. Scientists also would like to know. Things like fossil shells present no great problem as a rule, because the hard parts are external when the animal is alive and the outer appearance is actually preserved in the fossils.
Animals in which the skeleton is internal present great problems of restoration, and honest restorers admit that they often have to use considerable guessing. The general shape and contours of the body are fixed by the skeleton and by muscles attached to the skeleton, but surface features, which may give the animal its really characteristic look, are seldom restorable with any real probability of accuracy. The present often helps to interpret the past. An extinct animal presumably looked more or less like its living relatives, if it has any. This, however, may be quite equivocal. For example, extinct members of the horse family are usually restored to look somewhat like the most familiar living horses — domestic horses and their closest wild relatives. It is, however, possible and even probable that many extinct horses were striped like zebras. Others probably had patterns no longer present in any living members of the family. If lions and tigers were extinct they would be restored to look exactly alike. No living elephants have much hair and mammoths, which are extinct elephants, would doubtless be restored as hairless if we did not happen to know that they had thick, woolly coats. We know this only because mammoths are so recently extinct that prehistoric men drew pictures of them and that the hide and hair have actually been found in a few specimens. For older extinct animals we have no such clues.

The reconstruction of a fossilized animal’s external appearance is considered necessary in order to______.

A．(A) satisfy popular curiosity

B．(B) answer scientific questions

C．(C) establish its activities

D．(D) determine its eating habits
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填空题

对刑满释放出狱人员合法权益的保护应纳入______的系统工程。
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