2．佳句

(1)More and more people come to realize the importance of regular physical exercise.

(2)We must struggle against our own laziness and stay with our training, rain or shine.(风雨无阻)

(3)Maintaining our health is very important. No one should trifle with his health.

(4)Physical exercise increases the appetite and favors digestion. It increases the circulation of the blood. Physical exercise can develop one’s self-confidence, judgment, and a strong will.

(5)Early to bed and early to rise, makes a man healthy, wealthy, and wise. (Benjamin Franklin)

(6)A sound mind is in a sound body.

(7)Taking part in sports and games will keep us fit and healthy.

(8)Proper sports activities help us strengthen our physical body and build our sense of competition and cooperation.

(9)Research shows that getting plenty of exercise makes heart beat faster and lung work harder, thus strengthening the heart and reducing the chance of heart attack, and helping to lower blood pressure.

(10)For those who work with their brains most of the day, the practice of sport is especially useful.

(11)Once a habit is formed, it is difficult and sometimes impossible to shake it off.

(12)It is easier to fall into bad habits than to get into good ones.

(13)Many successful men declare that they own much of their prosperity to the formation of certain good habits in early life, such as punctuality, early rising, honesty, and thoroughness.

(14)The eating habits of Chinese people have changed dramatically in the past decade.

(15)People begin to eat less grain, but more fruit and vegetables. Fresh fruit and vegetables are rich in carbohydrate(碳水化合物), vitamins and minerals.

(16)People now pay much attention to nutrition, so they choose to eat meat and drink milk.

(17)Fish and chicken contain more protein and less fat.

(18)The issue of re-employment is of vital importance to the country’s reform, development and stability.

(19)Some statistics indicate that trained workers are three times as likely to be re-employed as those who haven’t participated in any training programs.

(20)A re-employment-oriented training network should be formed immediately to cover as many laid-off workers as possible to enhance their abilities to face challenges.

(21)Its no shame to earn bread with one’s own hands no matter what kind of work he does, but it’s a disgrace to idle along with folded arms and wait until financial aid arrives.

(22)China is undergoing a period of transition from a socialist planned economy to a socialist market economy.

(23)The government is supposed to issue more beneficial policies to help the lay-offs out of difficulty.

(24)Everyone desires and pursues happiness. But happiness means different things to different people.

(25)I’ll be happy if I can realize my value in my future career and make some contribution to the development of our country. (26)Happiness also means being on good term with my colleagues and friends.

(27)The only ones who will be really happy are those who will have sought and found how to serve the people.

(28)Happiness lies first of all in health.

(29)Happiness is always abounded from hard work.

(30)Happiness consists in contentment.

(31)Every generation has its own view of life and value system, which results from its living circumstances.

(32)As long as different generations can understand each other and avoid foisting their own views and values on others, there will be fewer conflicts and more harmony between the generations.

(33)The old assume that they know best, but it is only a matter of experience.

(34)The young know how to enjoy work and leisure and not to be inhibited.(约束)

(35)Children often complain that their parents cannot understand them, while parents feel sorry that their children seldom show them proper respect and obedience.

(36)Children should respect their parents and be aware that what their parents do is for their good.

(37)As reformation goes deeper in China, civil servants no longer have “iron rice bowls” as they used to have in many places.

(38)Whatever we do, we should do in our power.(尽力而为)

(39)To make a wise choice of occupations, two important things should be taken into consid-eration. One is the interest, the other, the demand of the people and society.

(40)It is definitely true that behind every daily activity lies a motive. As a matter of fact, we can achieve nothing without a realistic goal.

(41)Everybody should have a goal in his life, because aimless life wastes our energy and time.

(42)Whatever goal you might have, the primarily important thing you should have in life is health.

(43)It is universally true that everyone needs good health. With our society becoming more competitive, it is important to stay healthy.

(44)For one thing, people with good health can do work with full energy and their excellence in work in turn contributes to their health and happiness. For another, an unhealthy person is seldom able to show interest in everything around him and therefore he loses many opportunities to achieve success.

(45)Generally speaking, those who have good motives do their duties well. For example, parents work hard from early morning till late at night because they want to support their families. Students study diligently either to rank top in their class or to be prepared for their future success.

(46)On the other hand, people do things evil because they have evil goals. For instance, the motive for getting money without hard work makes a pickpocket steal a purse. To fulfill his evil desire, a robber grab others’ belongings, and a murderer can kill an innocent person. These wicked motives are the kind that people need to get rid of because they hurt others.

(47)It is often easier to have ideals than to carry them out.

(48)Effort and persistence are necessary for the realization of ideals.

(49)Once the goal is determined, we must create opportunities to reach it. Work hard and be patient after your goal is set. Try your best to make your life happen in the way you wish to see it. It is those who set the right goal and stick to it long enough will finally achieve it.

(50)Living without an aim is like sailing without a compass. When ideals are gone you may still exist, but you have ceased to live.(虽生犹死)

必背佳句：

•　　 1. It must be pointed out that it is one of our basic State policies to control population growth while raising the quality of the population. 一定要指出的是国家基本政策之一是在提高人口质量的同时控制人口增长。

•　　 2. It must be kept in mind that there is no secret of success but hard work. 一定要记住的是成功的秘密是努力的工作。

•　　 3. It can be seen from this that there is no difficulty in the world we cannot overcome.从这里可看出，世上没有克服不了的困难。

•　　 5. As is known to us, knowledge is power.众所周知，知识就是力量。

•　　 6. It is a common saying that where there is a will ,there is a way.俗话说，有志者，事竟成。

•　　 7. It is hard to imagine how Edison managed to work twenty hours each day.很难想象爱迪生每天是怎样工作20小时的。

•　　 8. It‘s hard to say whether the plan is practical or not.这个计划是否实际很难说。

•　　 9. There is no doubt that you will be helped by others if you have any difficulties.毫无疑问，你有困难时，会得到别人的帮助。

•　　 10. To tell the truth , many mistakes we made could have been avoided.老实说我们所犯的许多错误本来都能够避免的。

•　　 11. As we know, it was not until recently that the problem was solved.正如我们指的的一样，直到最近，这个问题才被解决。

•　　 13. All this shows that nothing can prevent us from reaching our aims.这显示了没有事情能够阻挡我们实现目标。

•　　 14. As far as we know, it took him more than a year to write the book.到目前为止我们所知道的是，他用了10年的时间来写这本书。

•　　 15. It has been proved that his theory is right.已经证明，他的理论是对的。

•　　 17. To be frank, whether you like it or not, you have no other choice.老实说，不论你喜不喜欢，你别无选择。

•　　 19. We will be successful as long as we insist on working hard.只要我们坚持努力工作，我们会成功的。

•　　 21. It is true that we must make our greater efforts; otherwise we cannot catch up with the developed countries.是真的，我们要作出更大的努力，不然/否则，我们不能赶上发达国家。

•　　 22. I take it for granted that they will support this idea.我认为他们会支持这个提议是理所当然的。

•　　 24. In a certain sense, a successful scientist is a person who is never satisfied with what he has achieved.在某种情况下，一个成功的科学家就是一个绝不满足于自己已取得的成就的人。

•　　 25. There is no denying the fact that the new management method has greatly increased the production. 不可否认的事实是，新的管理方法已经极大提高了产量。

•　　 26. Upon / On hearing the unexpected news, he was so surprised that he couldn‘t say a word. 一听到这个出乎意料的消息，他惊讶到说不出话来。

•　　 27, As the saying goes, nothing in the world is difficult for one who sets his mind to it.俗话说，世上无难事，只怕有心人。

•　　 28. Noting can prevent us from realizing the four modernizations.没有事情可以阻止我们实现四个现代化。

•　　 29. Now in China, more and more families can afford to buy high-grade goods, such as washing machines, TV sets, video-recorders.在今时今日的中国，越来越多的家庭有能力买高档次的货物，例如洗衣机、电视机和录像机。

•　　 30. No matter how difficult English may be, you should do your best to learn it.不管英语有多么难，你都应该尽你最大的努力来学它。

•　　 31. The number of visitors has increased year by year. As a result of　 Reform and Open Policy, the number of them has been increasing greatly each year.游客的数量逐年递增，由于改革开发政策，游客的数量已经每年大幅增加。

•　　 32. You will forget your tiredness and build up your health.你会忘记疲劳，建造健康。

•　　 33. But sometimes　 traveling　 is not an enjoyable thing, for example, the weather can be changeable. You may be caught in the rain and may catch a cold while travelling. 但有时候，旅游不一定是一件令人享受的事，举个例子，天气多变。你有可能在旅程中被雨淋或着凉感冒。

•　　 34. The worst thing is that you may have your money stolen and you may have an injury. All these are terrible things which can happen to a tourist.最糟糕的事情事你的钱可能被偷或者你也能发生意外。所有这些事情都是有可能发生在你身上的。

•　　 35. you must be careful everywhere and try to avoid accidents.你必须要处处小心，尽量去避免意外。

•　　 36．Our factories will try our best to meet the requirements of consumers.我们的工厂们会尽最大努力来满足顾客们的要求。

•　　 37.Last Sunday, our class organized some volunteer’s activities, in which all of us took an active part.上个星期天，我们班组织一些志愿活动，我们所有人都参加了。

•　　 38.These volunteer’s activities can help us to gain some social experience and make good sense of our personal values as well. 这些义务活动能帮我们获取一些社会经验同时也能帮我们意识到自身的价值。

•　　 39. Group three helped to raise money in the street for the “Hope Project”, so that more children in poor areas can afford their schooling.第三组帮忙到街上集款给“希望工程”，以至于有更多的贫困地区的孩子们能读得起书。

•　　 40. Compared with the traditional cards, electronic cards are more interesting and lively.与传统的卡片相比，电子卡更有趣和生动。

•　　 41. In Guangming Middle School, every Sunday afternoon from 2:30 p.m.to 4:30 p.m., there is an English corner. 在光明中学，每周日下午从2.30到4.30，有一个英语角。

•　　 42. Apart from students from Guangming Middle school, the attendants include students from other middle schools or colleges, even doctors and engineers.除了光明中学的学生之外，参加的人还有其他中学的学生或大学的学生，甚至还有医生和工程师们。

•　　 43.All the teachers and the students are expected to be present at the art festival. There is sure to be a lot of fun. Do come and join us.所有的老师和学生都要出席学校的艺术节。到时肯定很好玩。请务必到场加入我们。

•　　 44.Although Li Hua was the last one to cross the finishing line, he won the “Courage Cup” with honor. I was deeply touched by Li Hua’s great determination. 虽然李华是最后一个冲线的人，但他赢得了勇气杯。我被李华的坚强意志深深地感动了。

高考英语作文常用句型及短语

1．好词

(1)Population explosion/baby boom　　　　 (2)Population census

(3)Birth/death rate　　　　　　　　　　　 (4)Family planning

(5)Suicide bombing　　　　　　　　　　　 (6)Dropout students

(7)Eliminate illiteracy　　　　 　　　　　　　(8)Help those in distress and aid those in peril　　

　(9)Poverty-stricken areas　　　　　　　　 (10)Remote and mountain areas

(11)Shake off poverty and set out on a road to prosperity

(12)Brain drain　　　　　　　　　 　　　(13)Corruption phenomenon

(14)Offer and take bribes　　　　　　　　 (15)Embezzle pubic funds

(16)Counterfeit certificates　　　　　　　 (17)Counterfeit currency/ID card

(18)Credit card fraud　　　　　　　　　　 (19)Fake commodities

(20)Vending machine　　　　　　　　　　 (21)Generation gap

(22)Couch potato　　　　　　　　　　　 (23)Keep a concubine/mistress

(24)Casino/gambling　　　　　　　　　　 (25)Go on a diet

(26)Well-balanced meals

(27)Enjoy banquets using public funds　　　 (28)Bad construction projects

(29)Suspend/discontinue a project

(30)Welfare-oriented public housing distribution system

(31)High-rise apartment building

(32)Down payment (by monthly installments)

(33)Juvenile delinquency　　　　　　　　　　　　 (34)Laid-off workers

(35)Trial-period

(36)On-the-job training/professional training

(37)Reemployment project　　　　　　　　　　　　 (38)U.N. Security Council

(39)Government-funded personnel studying abroad

(40)Racial/sex discrimination　　　　　　　　　　　 (41)Traffic jam/road congestion

(42)Drunk driving　　　　　　 　　　　　　　　　　(43)Violation of traffic regulations

(44)Hit-and-run accident　　　　　　　　　　　　　 (45)Rush/peak hour

(46)Security personnel　　　　　　　　　　　　　 (47)Spokesman for news release

(48)Press conference　　　　　　　　　　　　　　 (49)Globalization

(50)Intellectual property right　　　　　　　　　　　　 (51)Infringement on the patent right

(52)Bid for the Olympic Games　　　　　　　　　　 (53)Sports lottery

(54)Live broadcast　　　　　　　　　　　　　　　 (55)Frozen food

(56)Preservation of cultural relics　　　　　　　　　　　

(57)Change from temporary to regular worker

(58)Body-building exercises　　　　　　　　　　　 (59)Bungee jumping

(60)Permanent residence certificate　　　　　　　　　 (61)Euthanasia

(62)Drug addition　　　　　　　 　　　　　　　　　(63)AIDS disease

(64)Campus violence　　　　　　　　　　　　　　 (65)Child abuse

(66)Kidnap/blackmail

2．佳句

(1)Pollution is, in fact, threatening our existence.

(2)Villagers are as a rule healthier than people who live in towns. One reason for this is that country air is fresher than the air in smoky cities.

(3)Most of the noise in the city comes from traffic, factories and construction sites.

(4)There is an increasingly loud voice from the public calling for firm action against pollution.

(5)Scientists have warned that unless effective solutions are worked out, the problem of pollution will eventually get out of hand.

(6)Many factories discharge enormous harmful chemicals into air and rivers every day.

(7)To kill insects, farmers use a large amount of insecticides so as to have a bumper harvest. As a result, they pollute the air, water and land.

(8)The gas from car engines is very poisonous. It is a main source of air pollution in cities.

(9)Deforestation result in the increase of global temperature and unpleasant change of climate.

(10)The earth is our home and we have the duty to take care of it for ourselves and for our later generation.

(11)Beijing now has become one of the most polluted cities in the world. Among the 10 cities with the worst air pollution in the world last year, Beijing was the third most polluted city.

(12)If the city had not plagued by poor quality air, the annual death rate in Beijing would have been 4 percent lower in past three years, according to the research done jointly by Beijing Medical University and Harvard University.

(13)Beijing consumes 2.8 million tons of coal each year, of which 75 percent are low-grade coal.

(14)To make room for those buildings, trees are being cut down, grass plots occupied. Cities look like nothing but grey concrete woods.

(15)Modern cities should be expanding with a certain percentage for green-plant areas.

(16)We must begin now to protect our only earth.

(17)Almost everything we use daily comes from the land directly or indirectly.

(18)But the limited land is decreasing at a surprising speed.

(19)The nature has given us so much, but looks at what we have done in return for its generosity.

(20)The ozone layer is our natural shield in the sky, but this shield is in great danger due to human activities. To protect the ozone layer is to protect us.

(21)Depleting the ozone layer allows more UV-B to reach the earth, which means more skin cancers, more eye cataracts(白内障)，weakened immune systems, and reduced plant yield.

(22)The nature has been irritated, which results in the shortage of valuable resources, air pollution, countless acres of lands becoming deserts, etc.

(23)Fortunately, many countries have got to know the seriousness of the problem and take active measures to preserve the environment.

(24)We sincerely hope that all the people in the world will take part in the campaign and join hands to protect the nature.

(25)Besides, trees and lawns in our cities will in the long run help improve the local climate.

(26)The government should also spend more money on the preservation of current lawns and cultivation of new lawns.

(27)Chinese government has taken strict measures to stop careless and indiscriminate felling of trees.

(28)Therefore, governments are playing the most important role in environmental protection today.

(29)The government should let people full realize the importance of environmental protection through education.

(30)Cars should be equipped with special devices to reduce auto emission.

(31)Farmers can utilize modern biological techniques instead of chemical fertilizers to grow crops.

(32)Desert can be turned into cultivated land by a certain kind of shrub.

(33)Recycling of paper, steel and plastics is important for helping protect our environment.

(34)The city was beautified on a tremendous scale.

(35)Energy crisis will threaten our existence because of the people’s extravagance and the rapid growth of he population.

(36)It is imperative that we investigate and develop new sources of energy immediately.

(37)With the expansion of industrialization, requirement for energy all over the world is on the rise.

(38)We have a great need for energy because of the rapid growth of our economy.

(39)These rich energy resources are being used up rapidly.

(40)To avoid energy crisis in the future, we should take some actions.

(41)The overuse of energy has influenced ecological balance.

(42)The development of modern life is placing and ever increasing demand for electricity.

(43)If there were no energy, our world would be in a state of confusion.

(44)Protecting endangered species helps protect a healthy environment. Endangered species are nature’s early warning system for pollution and environmental degradation that may someday affect human health.(濒危物种是大自然的关于污染和环境恶化的早期警戒系统，污染和环境恶化有朝一日会影响人类的健康。)

(45)Protecting endangered species saves a part of nature for our children and grandchildren to enjoy.

(46)Protecting endangered species helps protect sustainable economies and a good quality of life. Endangered species of fish, wildlife and plants are of ecological, educational, historical, recreational and scientific value to the nations and their people.

(47)All of the living creatures, including humans, are part of a complex, delicately balanced network called the biosphere.

(48)No creature exists in isolation. The removal of a single species can set off a chain reaction affecting many others.

(49)It is estimated that about 12,000 Tibetan antelopes are killed each year. If this trend continues, this species will be extinct in two decades.

(50)A healthy environment for wildlife contributes to healthy environment for people, today and tomorrow.

(二)社会生活问题

1．好词

(1)Environmental/water/air/noise pollution　　　　　 (2)Resources exhaustion

(3)Wildlife extinction　　　　　　　　　　　　　 (4)Endangered species

(5)Natural habitat　　　　　　　　　　　　　　　 (6)Reserve areas

(7)Sand/dust storm　　　　　　　　　　　　　　 (8)Clear-cutting/deforestation

(9)Over fishing　　　　　　　　　　　　　　　　 (10)Overgrazing

(11)Soil erosion　　　　　　　　　　　　　　　 (12)Water and soil conservation

(13)Desertification　　　　　　　　　　　　　　 (14)Natural disaster

(15)Flood threat　　　　　　　　　　　　　　　 (16)Fire hazard

(17)Storm/tempest　　　　　　　　　　　　　　 (18)Blizzard

(19)Drought　　　　　　　　　　　　　　　　　 (20)Famine

(21)Disruption of ecological balance(the balance of ecosystem)

(22)Eruption of volcanoes　　　　　　　　　　　 (23)Earthquake

(24)Avalanche　　　　　　　　　　　　　　　　 (25)Landslide

(26)Hurricane　　　　　　　　　　　　　　　　 (27)Shortage of water resources

(28)Shortage of fresh water　　　　　　　　　　　 (29)Harmful chemicals

(30)Poisonous/toxic gases　　　　　　　　　　　 (31)Carbon monoxide

(32)Urban smog　　　　　　　　　　　　　　　 (33)Industrial waste

(34)Hazardous nuclear waste　　　　　　　　　　 (35)Radioactive pollutants

(36)Waste gas sent off from automobiles　　　　　　 (37)Car emission standard

(38)Non-biodegradable material　　　　　　　　　 (39)Throw-away lunchbox

(40)Recyclable product　　　　　　　　　　　　 (41)Renewable resources

(42)Conserve natural resources　　　　　　　　　 (43)Plastic bags

(44)Greenhouse effect　　　　　　　　　　　　　 (45)Global warming

(46)Acid rain　　　　　　　　　　　　　　　　 (47)Energy crisis

(48)Oil leakage　　　　　　　　　　　　　　　　 (49)Environment awareness

(50)Pest rampancy　　　　　　　　　　　　　　 (51)Rescue and relief work

(52)Garbage disposal　　　　　

3．等比数列的性质

①等比数列任意两项间的关系：如果是等比数列的第项，是等差数列的第项，且，公比为，则有；

②对于等比数列，若，则，也就是：，如图所示：。

③若数列是等比数列，是其前n项的和，，那么，，成等比数列.

如下图所示：

2．等比数列的判定方法

①定义法：对于数列，若，则数列是等比数列；

②等比中项：对于数列，若，则数列是等比数列.

1．等比数列的知识要点(可类比等差数列学习)

(1)掌握等比数列定义＝q(常数)(nN)，同样是证明一个数列是等比数列的依据，也可由a_n·a_n+2＝来判断；

(2)等比数列的通项公式为a_n＝a₁·q^n－1；

(3)对于G 是a、b 的等差中项，则G²＝ab，G＝±；

(4)特别要注意等比数列前n 项和公式应分为q＝1与q≠1两类，当q＝1时，S_n＝na₁，当q≠1时，S_n＝，S_n＝。

题型1：等比数列的概念

例1．“公差为0的等差数列是等比数列”；“公比为的等比数列一定是递减数列”；“a,b,c三数成等比数列的充要条件是b²=ac”；“a,b,c三数成等差数列的充要条件是2b=a+c”，以上四个命题中，正确的有(　　 )

A．1个　　　　 　　 　　 B．2个　　　　　 　　　 　　 C．3个　　　　　　　 　 D．4个

解析：四个命题中只有最后一个是真命题。

命题1中未考虑各项都为0的等差数列不是等比数列；

命题2中可知a_n+1=a_n×，a_n+1<a_n未必成立，当首项a₁<0时，a_n<0，则a_n>a_n，即a_n+1>a_n，此时该数列为递增数列；

命题3中，若a=b=0，c∈R，此时有，但数列a,b,c不是等比数列，所以应是必要而不充分条件，若将条件改为b=，则成为不必要也不充分条件。

点评：该题通过一些选择题的形式考察了有关等比数列的一些重要结论，为此我们要注意一些有关等差数列、等比数列的重要结论。

例2．命题1：若数列{a_n}的前n项和S_n=aⁿ+b(a≠1)，则数列{a_n}是等比数列；

命题2：若数列{a_n}的前n项和S_n=an²+bn+c(a≠0)，则数列{a_n}是等差数列；

命题3：若数列{a_n}的前n项和S_n=na－n，则数列{a_n}既是等差数列，又是等比数列；上述三个命题中，真命题有(　　 )

A．0个　　　　　　　　 　　 B．1个　　　　　　　　 　　 C．2个　　　　　　　　 　 D．3个

解析： 由命题1得，a₁=a+b，当n≥2时，a_n=S_n－S_n－1=(a－1)·a^n－1。若{a_n}是等比数列，则=a，即=a，所以只有当b=－1且a≠0时，此数列才是等比数列。

由命题2得，a₁=a+b+c，当n≥2时，a_n=S_n－S_n－1=2na+b－a，若{a_n}是等差数列，则a₂－a₁=2a，即2a－c=2a，所以只有当c=0时，数列{a_n}才是等差数列。

由命题3得，a₁=a－1，当n≥2时，a_n=S_n－S_n－1=a－1，显然{a_n}是一个常数列，即公差为0的等差数列，因此只有当a－1≠0；即a≠1时数列{a_n}才又是等比数列。

点评：等比数列中通项与求和公式间有很大的联系，上述三个命题均涉及到S_n与a_n的关系，它们是a_n=，正确判断数列{a_n}是等差数列或等比数列，都必须用上述关系式，尤其注意首项与其他各项的关系。上述三个命题都不是真命题，选择A。

题型2：等比数列的判定

例3．已知等比数列中，则其前3项的和的取值范围是(D )

　(A)　　　　　　　　 (B)　

　(C)　　　　　　　　　 (D)

[解1]：∵等比数列中 ∴当公比为1时，， ；

　　　　 当公比为时，， 从而淘汰(A)(B)(C)

故选D；

[解2]：∵等比数列中 ∴

　 ∴当公比时，；

　 当公比时，　

∴　 故选D；

[考点]：此题重点考察等比数列前项和的意义，等比数列的通项公式，以及均值不等式的应用；

[突破]：特殊数列入手淘汰；重视等比数列的通项公式，前项和，以及均值不等式的应用，特别是均值不等式使用的条件；

点评：本题主要考查等比数列的概念和基本性质，推理和运算能力。

例4．(2009浙江文)设为数列的前项和，，，其中是常数．

　 (I) 求及；

　 (II)若对于任意的，，，成等比数列，求的值．

解(Ⅰ)当，

()

　经验，()式成立，　　　

(Ⅱ)成等比数列，，

即，整理得：，

对任意的成立，　　　　　

题型3：等比数列的通项公式及应用

例5．一个等比数列有三项，如果把第二项加上4，那么所得的三项就成为等差数列，如果再把这个等差数列的第三项加上32，那么所得的三项又成为等比数列，求原来的等比数列.

解析：设所求的等比数列为a，aq，aq²；

则2(aq+4)=a+aq²，且(aq+4)²=a(aq²+32)；

解得a=2，q=3或a=，q=－5；

故所求的等比数列为2，6,18或，－，。

点评：第一种解法利用等比数列的基本量，先求公比，后求其它量，这是解等差数列、等比数列的常用方法，其优点是思路简单、实用，缺点是有时计算较繁。

例6．(2009山东卷文)等比数列{}的前n项和为， 已知对任意的　 ，点，均在函数且均为常数)的图像上.　　　

(1)求r的值；　　　

(11)当b=2时，记　 　　求数列的前项和

解:因为对任意的,点，均在函数且均为常数)的图像上.所以得,

当时,,　　　

当时,,

又因为{}为等比数列,　 所以,　 公比为,　　 所以

(2)当b=2时，,　　

则

　　　

相减,得

所以

[命题立意]:本题主要考查了等比数列的定义,通项公式,以及已知求的基本题型,并运用错位相减法求出一等比数列与一等差数列对应项乘积所得新数列的前项和.

例7．(1)(2009安徽卷文)已知数列{} 的前n项和，数列{}的前n项和

(Ⅰ)求数列{}与{}的通项公式；

(Ⅱ)设，证明：当且仅当n≥3时，＜ 　　　　

[思路]由可求出，这是数列中求通项的常用方法之一，在求出后，进而得到，接下来用作差法来比较大小，这也是一常用方法.

[解析](1)由于

当时,

又当时

数列项与等比数列,其首项为1,公比为　　　　

(2)由(1)知

由即即

又时成立,即由于恒成立.　　　　

因此,当且仅当时,

点评：对于等比数列求和问题要先分清数列的通项公式，对应好首项和公比求出最终结果即可.

例8．(1)设{a_n}为等差数列，{b_n}为等比数列，a₁＝b₁＝1，a₂+a₄＝b₃，b₂b₄＝a₃．分别求出{a_n}及{b_n}的前10项的和S₁₀及T₁₀；

(2)在1与2之间插入n个正数a₁，a₂，a₃……，a_n，使这n+2个数成等比数列；又在1与2之间插入n个正数b₁，b₂，b₃，……，b_n，使这n+2个数成等差数列.记A_n＝a₁a₂a₃……a_n，B_n＝b₁+b₂+b₃+……+b_n.

(Ⅰ)求数列{A_n}和{B_n}的通项；

(Ⅱ)当n≥7时，比较A_n与B_n的大小，并证明你的结论。

(3)已知{a_n}是由非负整数组成的数列，满足a₁＝0，a₂＝3，

a_n+1a_n＝(a_n－1+2)(a_n－2+2)，n＝3，4，5，…．

(Ⅰ)求a₃；

(Ⅱ)证明a_n＝a_n－2+2，n＝3，4，5，…；

(Ⅲ)求{a_n}的通项公式及其前n项和S_n。

解析：(1)∵{a_n}为等差数列，{b_n}为等比数列，

∴a₂+a₄＝2a₃，b₂b₄＝b₃²．

已知a₂+a₄＝b₃，b₂b₄＝a₃，

∴b₃＝2a₃，a₃＝b₃²．

得 b₃＝2b₃²．

∵b₃≠0　 ∴b₃＝，a₃＝．

由a₁＝1，a₃＝知{a_n}的公差为d＝，

∴S₁₀＝10a₁+．

由b₁＝1，b₃＝知{b_n}的公比为q＝或q＝．

当q＝时，，

当q＝时，。

(2)(Ⅰ)设公比为q，公差为d，等比数列1，a₁，a₂，……，a_n，2，等差数列1，b₁，b₂，……，b_n，2。

则A₁＝a₁＝1·q　 A₂＝1·q·1·q²　 A₃＝1·q·1·q²·1·q³

又∵a_n+2＝1·qⁿ⁺¹＝2得qⁿ⁺¹＝2，

A_n＝q·q²…qⁿ＝q(n＝1，2，3…)

又∵b_n+2＝1+(n+1)d＝2　 ∴(n+1)d＝1

B₁＝b₁＝1+d　 B₂＝b₂+b₁＝1+d+1+2d　 B_n＝1+d+…+1+nd＝n

(Ⅱ)A_n＞B_n，当n≥7时

证明：当n＝7时，2^3．5＝8·＝A_n　 B_n＝×7，∴A_n＞B_n

设当n＝k时，A_n＞B_n，则当n＝k+1时，　　　　　　　

又∵A_k+1＝·　 且A_k＞B_k　 ∴A_k+1＞·k

∴A_k+1－B_k+1＞

又∵k＝8，9，10…　 ∴A_k+1－B_k+1＞0，综上所述，A_n＞B_n成立.

(3)(Ⅰ)解：由题设得a₃a₄＝10，且a₃、a₄均为非负整数，所以a₃的可能的值为1，2，5，10．

若a₃＝1，则a₄＝10，a₅＝，与题设矛盾．

若a₃＝5，则a₄＝2，a₅＝，与题设矛盾．

若a₃＝10，则a₄＝1，a₅＝60，a₆＝，与题设矛盾.

所以a₃＝2.

(Ⅱ)用数学归纳法证明：

①当n＝3，a₃＝a₁+2，等式成立；

②假设当n＝k(k≥3)时等式成立，即a_k＝a_k－2+2,由题设a_k+1a_k＝(a_k－1+2)·(a_k－2+2)，因为a_k＝a_k－2+2≠0，所以a_k+1＝a_k－1+2，

也就是说，当n＝k+1时，等式a_k+1＝a_k－1+2成立；

根据①和②，对于所有n≥3，有a_n+1=a_n－1+2。

(Ⅲ)解：由a_2k－1＝a_2(k－1)－1+2，a₁＝0，及a_2k＝a_2(k－1)+2，a₂＝3得a_2k－1＝2(k－1)，a_2k＝2k+1，k＝1，2，3，…，即a_n＝n+(－1)ⁿ，n＝1，2，3，…。

所以S_n＝

点评：本小题主要考查数列与等差数列前n项和等基础知识，以及准确表述，分析和解决问题的能力。

题型5：等比数列的性质

例9．(1)在各项都为正数的等比数列{a_n}中，首项a₁＝3，前三项和为21，则a₃+a₄+a₅＝(　　 　)

(A)33　　　　 (B)72　　　　 (C)84　　　　 (D)189

(2)(2000上海，12)在等差数列{a_n}中，若a₁₀＝0，则有等式a₁+a₂+…+a_n=a₁+a₂+…+a_19－n(n＜19，n∈N成立.类比上述性质，相应地：在等比数列{b_n}中，若b₉＝1，则有等式　　 成立.

解析：(1)答案：C；解：设等比数列{a_n}的公比为q(q>0)，由题意得:a₁+a₂+a₃=21,即3+3q+3q²=21,q²+q-6=0，求得q=2(q=－3舍去)，所以a₃+a₄+a₅=q²(a₁+a₂+a₃)=4故选C。

(2)答案：b₁b₂…b_n＝b₁b₂…b_17－n(n＜17，n∈N^*)；

解：在等差数列{a_n}中，由a₁₀＝0，得a₁+a₁₉＝a₂+a₁₈＝…＝a_n+a_20－n＝a_n+1+a_19－n＝2a₁₀＝0，

所以a₁+a₂+…+a_n+…+a₁₉＝0，即a₁+a₂+…+a_n＝－a₁₉－a₁₈－…－a_n+1，

又∵a₁＝－a₁₉，a₂＝－a₁₈，…，a_19－n＝－a_n+1

∴a₁+a₂+…+a_n＝－a₁₉－a₁₈－…－a_n+1＝a₁+a₂+…+a_19－n，

若a₉＝0，同理可得a₁+a₂+…+a_n＝a₁+a₂+a_17－n，

相应地等比数列{b_n}中，则可得：b₁b₂…b_n＝b₁b₂…b_17－n(n＜17，n∈N^*)。

点评：本题考查了等比数列的相关概念及其有关计算能力。

例10．(1)设首项为正数的等比数列，它的前n项和为80，前2n项和为6560，且前n项中数值最大的项为54，求此数列的首项和公比q。

(2)在和之间插入n个正数，使这个数依次成等比数列，求所插入的n个数之积。

(3)设等比数列{a_n}的各项均为正数，项数是偶数，它的所有项的和等于偶数项和的4倍，且第二项与第四项的积是第3项与第4项和的9倍，问数列{lga_n}的前多少项和最大？(lg2=0.　 3,lg3=0.4)

解析：(1)设等比数列{a_n}的前n项和为S_n，依题意设：a₁＞0，S_n=80 ，S_2n=6560。

　∵S_2n≠2S_n ，∴q≠1；

从而 =80，且=6560。

两式相除得1+qⁿ=82 ，即qⁿ=81。

∴a₁=q－1＞0 即q＞1,从而等比数列{a_n}为递增数列，故前n项中数值最大的项为第n项。

∴a₁q^n-1=54,从而(q－1)q^n-1=qⁿ-q^n-1=54。

∴q^n-1=81－54=27

∴q==3。

∴a₁=q－1=2

故此数列的首为2，公比为3。

(2)解法1：设插入的n个数为，且公比为q，

则

。

解法2：设插入的n个数为，

。

(3)解法一：　 设公比为q,项数为2m,m∈N^*，

依题意有：，

化简得，

设数列{lga_n}前n项和为S_n，

则S_n=lga₁+lga₁q²+…+lga₁q^n－1=lga₁ⁿ·q^{1+2+…+(n－1)}

=nlga₁+n(n－1)·lgq=n(2lg2+lg3)－n(n－1)lg3

=(－)·n²+(2lg2+lg3)·n

可见，当n=时，S_n最大，

而=5,故{lga_n}的前5项和最大，

解法二：　 接前，,于是lga_n=lg［108()^n－1］=lg108+(n－1)lg，

∴数列{lga_n}是以lg108为首项，以lg为公差的等差数列，

令lga_n≥0，得2lg2－(n－4)lg3≥0，

∴n≤=5.5，

由于n∈N^*,可见数列{lga_n}的前5项和最大。

点评：第一种解法利用等比数列的基本量，先求公比，后求其它量，这是解等差数列、等比数列的常用方法，其优点是思路简单、实用，缺点是有时计算较繁；第二种解法利用等比数列的性质，与“首末项等距”的两项积相等，这在解题中常用到。

题型6：等差、等比综合问题

例11．已知公比为的无穷等比数列各项的和为9，无穷等比数列各项的和为。

(Ⅰ)求数列的首项和公比；

(Ⅱ)对给定的，设是首项为，公差为的等差数列．求数列的前10项之和.

解析：(Ⅰ)依题意可知：，

(Ⅱ)由(Ⅰ)知,，所以数列的的首项为,公差，,即数列的前10项之和为155。

点评：对于出现等差、等比数列的综合问题，一定要区分开各自的公式，不要混淆。

4．等比数列前n项和公式

一般地，设等比数列的前n项和是，当时， 或；当q=1时，(错位相减法)。

说明：(1)和各已知三个可求第四个；(2)注意求和公式中是，通项公式中是不要混淆；(3)应用求和公式时，必要时应讨论的情况。

3．等比中项

如果在中间插入一个数，使成等比数列，那么叫做的等比中项(两个符号相同的非零实数，都有两个等比中项).