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.(風(fēng)雨無阻)

(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.(約束)

(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.(盡力而為)

(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.(雖生猶死)

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. 一定要指出的是國家基本政策之一是在提高人口質(zhì)量的同時控制人口增長。

•　　 2. It must be kept in mind that there is no secret of success but hard work. 一定要記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.從這里可看出，世上沒有克服不了的困難。

•　　 5. As is known to us, knowledge is power.眾所周知，知識就是力量。

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

•　　 7. It is hard to imagine how Edison managed to work twenty hours each day.很難想象愛迪生每天是怎樣工作20小時的。

•　　 8. It‘s hard to say whether the plan is practical or not.這個計劃是否實(shí)際很難說。

•　　 9. There is no doubt that you will be helped by others if you have any difficulties.毫無疑問，你有困難時，會得到別人的幫助。

•　　 10. To tell the truth , many mistakes we made could have been avoided.老實(shí)說我們所犯的許多錯誤本來都能夠避免的。

•　　 11. As we know, it was not until recently that the problem was solved.正如我們指的的一樣，直到最近，這個問題才被解決。

•　　 13. All this shows that nothing can prevent us from reaching our aims.這顯示了沒有事情能夠阻擋我們實(shí)現(xiàn)目標(biāo)。

•　　 14. As far as we know, it took him more than a year to write the book.到目前為止我們所知道的是，他用了10年的時間來寫這本書。

•　　 15. It has been proved that his theory is right.已經(jīng)證明，他的理論是對的。

•　　 17. To be frank, whether you like it or not, you have no other choice.老實(shí)說，不論你喜不喜歡，你別無選擇。

•　　 19. We will be successful as long as we insist on working hard.只要我們堅持努力工作，我們會成功的。

•　　 21. It is true that we must make our greater efforts; otherwise we cannot catch up with the developed countries.是真的，我們要作出更大的努力，不然/否則，我們不能趕上發(fā)達(dá)國家。

•　　 22. I take it for granted that they will support this idea.我認(rèn)為他們會支持這個提議是理所當(dāng)然的。

•　　 24. In a certain sense, a successful scientist is a person who is never satisfied with what he has achieved.在某種情況下，一個成功的科學(xué)家就是一個絕不滿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. 不可否認(rèn)的事實(shí)是，新的管理方法已經(jīng)極大提高了產(chǎn)量。

•　　 26. Upon / On hearing the unexpected news, he was so surprised that he couldn‘t say a word. 一聽到這個出乎意料的消息，他驚訝到說不出話來。

•　　 27, As the saying goes, nothing in the world is difficult for one who sets his mind to it.俗話說，世上無難事，只怕有心人。

•　　 28. Noting can prevent us from realizing the four modernizations.沒有事情可以阻止我們實(shí)現(xiàn)四個現(xiàn)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.在今時今日的中國，越來越多的家庭有能力買高檔次的貨物，例如洗衣機(jī)、電視機(jī)和錄像機(jī)。

•　　 30. No matter how difficult English may be, you should do your best to learn it.不管英語有多么難，你都應(yīng)該盡你最大的努力來學(xué)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.游客的數(shù)量逐年遞增，由于改革開發(fā)政策，游客的數(shù)量已經(jīng)每年大幅增加。

•　　 32. You will forget your tiredness and build up your health.你會忘記疲勞，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. 但有時候，旅游不一定是一件令人享受的事，舉個例子，天氣多變。你有可能在旅程中被雨淋或著涼感冒。

•　　 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錢可能被偷或者你也能發(fā)生意外。所有這些事情都是有可能發(fā)生在你身上的。

•　　 35. you must be careful everywhere and try to avoid accidents.你必須要處處小心，盡量去避免意外。

•　　 36．Our factories will try our best to meet the requirements of consumers.我們的工廠們會盡最大努力來滿足顧客們的要求。

•　　 37.Last Sunday, our class organized some volunteer’s activities, in which all of us took an active part.上個星期天，我們班組織一些志愿活動，我們所有人都參加了。

•　　 38.These volunteer’s activities can help us to gain some social experience and make good sense of our personal values as well. 這些義務(wù)活動能幫我們獲取一些社會經(jīng)驗(yàn)同時也能幫我們意識到自身的價值。

•　　 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.第三組幫忙到街上集款給“希望工程”，以至于有更多的貧困地區(qū)的孩子們能讀得起書。

•　　 40. Compared with the traditional cards, electronic cards are more interesting and lively.與傳統(tǒng)的卡片相比，電子卡更有趣和生動。

•　　 41. In Guangming Middle School, every Sunday afternoon from 2:30 p.m.to 4:30 p.m., there is an English corner. 在光明中學(xué)，每周日下午從2.30到4.30，有一個英語角。

•　　 42. Apart from students from Guangming Middle school, the attendants include students from other middle schools or colleges, even doctors and engineers.除了光明中學(xué)的學(xué)生之外，參加的人還有其他中學(xué)的學(xué)生或大學(xué)的學(xué)生，甚至還有醫(yī)生和工程師們。

•　　 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.所有的老師和學(xué)生都要出席學(xué)校的藝術(shù)節(jié)。到時肯定很好玩。請務(wù)必到場加入我們。

•　　 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. 雖然李華是最后一個沖線的人，但他贏得了勇氣杯。我被李華的堅強(qiáng)意志深深地感動了。

高考英語作文常用句型及短語

1．好詞

(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(白內(nèi)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.(瀕危物種是大自然的關(guān)于污染和環(huán)境惡化的早期警戒系統(tǒng)，污染和環(huán)境惡化有朝一日會影響人類的健康。)

(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)社會生活問題

1．好詞

(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．等比數(shù)列的性質(zhì)

①等比數(shù)列任意兩項(xiàng)間的關(guān)系：如果是等比數(shù)列的第項(xiàng)，是等差數(shù)列的第項(xiàng)，且，公比為，則有；

②對于等比數(shù)列，若，則，也就是：，如圖所示：。

③若數(shù)列是等比數(shù)列，是其前n項(xiàng)的和，，那么，，成等比數(shù)列.

如下圖所示：

2．等比數(shù)列的判定方法

①定義法：對于數(shù)列，若，則數(shù)列是等比數(shù)列；

②等比中項(xiàng)：對于數(shù)列，若，則數(shù)列是等比數(shù)列.

1．等比數(shù)列的知識要點(diǎn)(可類比等差數(shù)列學(xué)習(xí))

(1)掌握等比數(shù)列定義＝q(常數(shù))(nN)，同樣是證明一個數(shù)列是等比數(shù)列的依據(jù)，也可由a_n·a_n+2＝來判斷；

(2)等比數(shù)列的通項(xiàng)公式為a_n＝a₁·q^n－1；

(3)對于G 是a、b 的等差中項(xiàng)，則G²＝ab，G＝±；

(4)特別要注意等比數(shù)列前n 項(xiàng)和公式應(yīng)分為q＝1與q≠1兩類，當(dāng)q＝1時，S_n＝na₁，當(dāng)q≠1時，S_n＝，S_n＝。

題型1：等比數(shù)列的概念

例1．“公差為0的等差數(shù)列是等比數(shù)列”；“公比為的等比數(shù)列一定是遞減數(shù)列”；“a,b,c三數(shù)成等比數(shù)列的充要條件是b²=ac”；“a,b,c三數(shù)成等差數(shù)列的充要條件是2b=a+c”，以上四個命題中，正確的有(　　 )

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

解析：四個命題中只有最后一個是真命題。

命題1中未考慮各項(xiàng)都為0的等差數(shù)列不是等比數(shù)列；

命題2中可知a_n+1=a_n×，a_n+1<a_n未必成立，當(dāng)首項(xiàng)a₁<0時，a_n<0，則a_n>a_n，即a_n+1>a_n，此時該數(shù)列為遞增數(shù)列；

命題3中，若a=b=0，c∈R，此時有，但數(shù)列a,b,c不是等比數(shù)列，所以應(yīng)是必要而不充分條件，若將條件改為b=，則成為不必要也不充分條件。

點(diǎn)評：該題通過一些選擇題的形式考察了有關(guān)等比數(shù)列的一些重要結(jié)論，為此我們要注意一些有關(guān)等差數(shù)列、等比數(shù)列的重要結(jié)論。

例2．命題1：若數(shù)列{a_n}的前n項(xiàng)和S_n=aⁿ+b(a≠1)，則數(shù)列{a_n}是等比數(shù)列；

命題2：若數(shù)列{a_n}的前n項(xiàng)和S_n=an²+bn+c(a≠0)，則數(shù)列{a_n}是等差數(shù)列；

命題3：若數(shù)列{a_n}的前n項(xiàng)和S_n=na－n，則數(shù)列{a_n}既是等差數(shù)列，又是等比數(shù)列；上述三個命題中，真命題有(　　 )

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

解析： 由命題1得，a₁=a+b，當(dāng)n≥2時，a_n=S_n－S_n－1=(a－1)·a^n－1。若{a_n}是等比數(shù)列，則=a，即=a，所以只有當(dāng)b=－1且a≠0時，此數(shù)列才是等比數(shù)列。

由命題2得，a₁=a+b+c，當(dāng)n≥2時，a_n=S_n－S_n－1=2na+b－a，若{a_n}是等差數(shù)列，則a₂－a₁=2a，即2a－c=2a，所以只有當(dāng)c=0時，數(shù)列{a_n}才是等差數(shù)列。

由命題3得，a₁=a－1，當(dāng)n≥2時，a_n=S_n－S_n－1=a－1，顯然{a_n}是一個常數(shù)列，即公差為0的等差數(shù)列，因此只有當(dāng)a－1≠0；即a≠1時數(shù)列{a_n}才又是等比數(shù)列。

點(diǎn)評：等比數(shù)列中通項(xiàng)與求和公式間有很大的聯(lián)系，上述三個命題均涉及到S_n與a_n的關(guān)系，它們是a_n=，正確判斷數(shù)列{a_n}是等差數(shù)列或等比數(shù)列，都必須用上述關(guān)系式，尤其注意首項(xiàng)與其他各項(xiàng)的關(guān)系。上述三個命題都不是真命題，選擇A。

題型2：等比數(shù)列的判定

例3．已知等比數(shù)列中，則其前3項(xiàng)的和的取值范圍是(D )

　(A)　　　　　　　　 (B)　

　(C)　　　　　　　　　 (D)

[解1]：∵等比數(shù)列中 ∴當(dāng)公比為1時，， ；

　　　　 當(dāng)公比為時，， 從而淘汰(A)(B)(C)

故選D；

[解2]：∵等比數(shù)列中 ∴

　 ∴當(dāng)公比時，；

　 當(dāng)公比時，　

∴　 故選D；

[考點(diǎn)]：此題重點(diǎn)考察等比數(shù)列前項(xiàng)和的意義，等比數(shù)列的通項(xiàng)公式，以及均值不等式的應(yīng)用；

[突破]：特殊數(shù)列入手淘汰；重視等比數(shù)列的通項(xiàng)公式，前項(xiàng)和，以及均值不等式的應(yīng)用，特別是均值不等式使用的條件；

點(diǎn)評：本題主要考查等比數(shù)列的概念和基本性質(zhì)，推理和運(yùn)算能力。

例4．(2009浙江文)設(shè)為數(shù)列的前項(xiàng)和，，，其中是常數(shù)．

　 (I) 求及；

　 (II)若對于任意的，，，成等比數(shù)列，求的值．

解(Ⅰ)當(dāng)，

()

　經(jīng)驗(yàn)，()式成立，　　　

(Ⅱ)成等比數(shù)列，，

即，整理得：，

對任意的成立，　　　　　

題型3：等比數(shù)列的通項(xiàng)公式及應(yīng)用

例5．一個等比數(shù)列有三項(xiàng)，如果把第二項(xiàng)加上4，那么所得的三項(xiàng)就成為等差數(shù)列，如果再把這個等差數(shù)列的第三項(xiàng)加上32，那么所得的三項(xiàng)又成為等比數(shù)列，求原來的等比數(shù)列.

解析：設(shè)所求的等比數(shù)列為a，aq，aq²；

則2(aq+4)=a+aq²，且(aq+4)²=a(aq²+32)；

解得a=2，q=3或a=，q=－5；

故所求的等比數(shù)列為2，6,18或，－，。

點(diǎn)評：第一種解法利用等比數(shù)列的基本量，先求公比，后求其它量，這是解等差數(shù)列、等比數(shù)列的常用方法，其優(yōu)點(diǎn)是思路簡單、實(shí)用，缺點(diǎn)是有時計算較繁。

例6．(2009山東卷文)等比數(shù)列{}的前n項(xiàng)和為， 已知對任意的　 ，點(diǎn)，均在函數(shù)且均為常數(shù))的圖像上.　　　

(1)求r的值；　　　

(11)當(dāng)b=2時，記　 　　求數(shù)列的前項(xiàng)和

解:因?yàn)閷θ我獾?sub>,點(diǎn)，均在函數(shù)且均為常數(shù))的圖像上.所以得,

當(dāng)時,,　　　

當(dāng)時,,

又因?yàn)閧}為等比數(shù)列,　 所以,　 公比為,　　 所以

(2)當(dāng)b=2時，,　　

則

　　　

相減,得

所以

[命題立意]:本題主要考查了等比數(shù)列的定義,通項(xiàng)公式,以及已知求的基本題型,并運(yùn)用錯位相減法求出一等比數(shù)列與一等差數(shù)列對應(yīng)項(xiàng)乘積所得新數(shù)列的前項(xiàng)和.

例7．(1)(2009安徽卷文)已知數(shù)列{} 的前n項(xiàng)和，數(shù)列{}的前n項(xiàng)和

(Ⅰ)求數(shù)列{}與{}的通項(xiàng)公式；

(Ⅱ)設(shè)，證明：當(dāng)且僅當(dāng)n≥3時，＜ 　　　　

[思路]由可求出，這是數(shù)列中求通項(xiàng)的常用方法之一，在求出后，進(jìn)而得到，接下來用作差法來比較大小，這也是一常用方法.

[解析](1)由于

當(dāng)時,

又當(dāng)時

數(shù)列項(xiàng)與等比數(shù)列,其首項(xiàng)為1,公比為　　　　

(2)由(1)知

由即即

又時成立,即由于恒成立.　　　　

因此,當(dāng)且僅當(dāng)時,

點(diǎn)評：對于等比數(shù)列求和問題要先分清數(shù)列的通項(xiàng)公式，對應(yīng)好首項(xiàng)和公比求出最終結(jié)果即可.

例8．(1)設(shè){a_n}為等差數(shù)列，{b_n}為等比數(shù)列，a₁＝b₁＝1，a₂+a₄＝b₃，b₂b₄＝a₃．分別求出{a_n}及{b_n}的前10項(xiàng)的和S₁₀及T₁₀；

(2)在1與2之間插入n個正數(shù)a₁，a₂，a₃……，a_n，使這n+2個數(shù)成等比數(shù)列；又在1與2之間插入n個正數(shù)b₁，b₂，b₃，……，b_n，使這n+2個數(shù)成等差數(shù)列.記A_n＝a₁a₂a₃……a_n，B_n＝b₁+b₂+b₃+……+b_n.

(Ⅰ)求數(shù)列{A_n}和{B_n}的通項(xiàng)；

(Ⅱ)當(dāng)n≥7時，比較A_n與B_n的大小，并證明你的結(jié)論。

(3)已知{a_n}是由非負(fù)整數(shù)組成的數(shù)列，滿足a₁＝0，a₂＝3，

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

(Ⅰ)求a₃；

(Ⅱ)證明a_n＝a_n－2+2，n＝3，4，5，…；

(Ⅲ)求{a_n}的通項(xiàng)公式及其前n項(xiàng)和S_n。

解析：(1)∵{a_n}為等差數(shù)列，{b_n}為等比數(shù)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}的公差為d＝，

∴S₁₀＝10a₁+．

由b₁＝1，b₃＝知{b_n}的公比為q＝或q＝．

當(dāng)q＝時，，

當(dāng)q＝時，。

(2)(Ⅰ)設(shè)公比為q，公差為d，等比數(shù)列1，a₁，a₂，……，a_n，2，等差數(shù)列1，b₁，b₂，……，b_n，2。

則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，當(dāng)n≥7時

證明：當(dāng)n＝7時，2^3．5＝8·＝A_n　 B_n＝×7，∴A_n＞B_n

設(shè)當(dāng)n＝k時，A_n＞B_n，則當(dāng)n＝k+1時，　　　　　　　

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，綜上所述，A_n＞B_n成立.

(3)(Ⅰ)解：由題設(shè)得a₃a₄＝10，且a₃、a₄均為非負(fù)整數(shù)，所以a₃的可能的值為1，2，5，10．

若a₃＝1，則a₄＝10，a₅＝，與題設(shè)矛盾．

若a₃＝5，則a₄＝2，a₅＝，與題設(shè)矛盾．

若a₃＝10，則a₄＝1，a₅＝60，a₆＝，與題設(shè)矛盾.

所以a₃＝2.

(Ⅱ)用數(shù)學(xué)歸納法證明：

①當(dāng)n＝3，a₃＝a₁+2，等式成立；

②假設(shè)當(dāng)n＝k(k≥3)時等式成立，即a_k＝a_k－2+2,由題設(shè)a_k+1a_k＝(a_k－1+2)·(a_k－2+2)，因?yàn)?i>a_k＝a_k－2+2≠0，所以a_k+1＝a_k－1+2，

也就是說，當(dāng)n＝k+1時，等式a_k+1＝a_k－1+2成立；

根據(jù)①和②，對于所有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＝

點(diǎn)評：本小題主要考查數(shù)列與等差數(shù)列前n項(xiàng)和等基礎(chǔ)知識，以及準(zhǔn)確表述，分析和解決問題的能力。

題型5：等比數(shù)列的性質(zhì)

例9．(1)在各項(xiàng)都為正數(shù)的等比數(shù)列{a_n}中，首項(xiàng)a₁＝3，前三項(xiàng)和為21，則a₃+a₄+a₅＝(　　 　)

(A)33　　　　 (B)72　　　　 (C)84　　　　 (D)189

(2)(2000上海，12)在等差數(shù)列{a_n}中，若a₁₀＝0，則有等式a₁+a₂+…+a_n=a₁+a₂+…+a_19－n(n＜19，n∈N成立.類比上述性質(zhì)，相應(yīng)地：在等比數(shù)列{b_n}中，若b₉＝1，則有等式　　 成立.

解析：(1)答案：C；解：設(shè)等比數(shù)列{a_n}的公比為q(q>0)，由題意得: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故選C。

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

解：在等差數(shù)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，

相應(yīng)地等比數(shù)列{b_n}中，則可得：b₁b₂…b_n＝b₁b₂…b_17－n(n＜17，n∈N^*)。

點(diǎn)評：本題考查了等比數(shù)列的相關(guān)概念及其有關(guān)計算能力。

例10．(1)設(shè)首項(xiàng)為正數(shù)的等比數(shù)列，它的前n項(xiàng)和為80，前2n項(xiàng)和為6560，且前n項(xiàng)中數(shù)值最大的項(xiàng)為54，求此數(shù)列的首項(xiàng)和公比q。

(2)在和之間插入n個正數(shù)，使這個數(shù)依次成等比數(shù)列，求所插入的n個數(shù)之積。

(3)設(shè)等比數(shù)列{a_n}的各項(xiàng)均為正數(shù)，項(xiàng)數(shù)是偶數(shù)，它的所有項(xiàng)的和等于偶數(shù)項(xiàng)和的4倍，且第二項(xiàng)與第四項(xiàng)的積是第3項(xiàng)與第4項(xiàng)和的9倍，問數(shù)列{lga_n}的前多少項(xiàng)和最大？(lg2=0.　 3,lg3=0.4)

解析：(1)設(shè)等比數(shù)列{a_n}的前n項(xiàng)和為S_n，依題意設(shè)：a₁＞0，S_n=80 ，S_2n=6560。

　∵S_2n≠2S_n ，∴q≠1；

從而 =80，且=6560。

兩式相除得1+qⁿ=82 ，即qⁿ=81。

∴a₁=q－1＞0 即q＞1,從而等比數(shù)列{a_n}為遞增數(shù)列，故前n項(xiàng)中數(shù)值最大的項(xiàng)為第n項(xiàng)。

∴a₁q^n-1=54,從而(q－1)q^n-1=qⁿ-q^n-1=54。

∴q^n-1=81－54=27

∴q==3。

∴a₁=q－1=2

故此數(shù)列的首為2，公比為3。

(2)解法1：設(shè)插入的n個數(shù)為，且公比為q，

則

。

解法2：設(shè)插入的n個數(shù)為，

。

(3)解法一：　 設(shè)公比為q,項(xiàng)數(shù)為2m,m∈N^*，

依題意有：，

化簡得，

設(shè)數(shù)列{lga_n}前n項(xiàng)和為S_n，

則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

可見，當(dāng)n=時，S_n最大，

而=5,故{lga_n}的前5項(xiàng)和最大，

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

∴數(shù)列{lga_n}是以lg108為首項(xiàng)，以lg為公差的等差數(shù)列，

令lga_n≥0，得2lg2－(n－4)lg3≥0，

∴n≤=5.5，

由于n∈N^*,可見數(shù)列{lga_n}的前5項(xiàng)和最大。

點(diǎn)評：第一種解法利用等比數(shù)列的基本量，先求公比，后求其它量，這是解等差數(shù)列、等比數(shù)列的常用方法，其優(yōu)點(diǎn)是思路簡單、實(shí)用，缺點(diǎn)是有時計算較繁；第二種解法利用等比數(shù)列的性質(zhì)，與“首末項(xiàng)等距”的兩項(xiàng)積相等，這在解題中常用到。

題型6：等差、等比綜合問題

例11．已知公比為的無窮等比數(shù)列各項(xiàng)的和為9，無窮等比數(shù)列各項(xiàng)的和為。

(Ⅰ)求數(shù)列的首項(xiàng)和公比；

(Ⅱ)對給定的，設(shè)是首項(xiàng)為，公差為的等差數(shù)列．求數(shù)列的前10項(xiàng)之和.

解析：(Ⅰ)依題意可知：，

(Ⅱ)由(Ⅰ)知,，所以數(shù)列的的首項(xiàng)為,公差，,即數(shù)列的前10項(xiàng)之和為155。

點(diǎn)評：對于出現(xiàn)等差、等比數(shù)列的綜合問題，一定要區(qū)分開各自的公式，不要混淆。

4．等比數(shù)列前n項(xiàng)和公式

一般地，設(shè)等比數(shù)列的前n項(xiàng)和是，當(dāng)時， 或；當(dāng)q=1時，(錯位相減法)。

說明：(1)和各已知三個可求第四個；(2)注意求和公式中是，通項(xiàng)公式中是不要混淆；(3)應(yīng)用求和公式時，必要時應(yīng)討論的情況。

3．等比中項(xiàng)

如果在中間插入一個數(shù)，使成等比數(shù)列，那么叫做的等比中項(xiàng)(兩個符號相同的非零實(shí)數(shù)，都有兩個等比中項(xiàng)).