题目

用数学归纳法证明凸n边形的对角线的条数f(n)=n(n-3)(n≥3). 答案：分析：证明几何问题,一定要弄清从n=k到n=k+1时,新增加的量是多少.一般地,证明第二步时,常用的方法是加一法,即在原来k的基础上,再增加1个,也可以从k+1个中分出1个来,剩下的k个利用假设.证明：(1)当n=3时,f(3)=0,因为三角形没有对角线,所以命题成立.(2)假设n=k时(k≥3),f(k)=k(k-3)成立,当n=k+1时,凸k+1边形由原来的Most people feel lonely sometimes，but it usually only lasts between a few minutes and a few hours.This kind of loneliness is not serious.In fact，it is quite normal.For some people，though，loneliness can last for years.Now researchers say there are three different types of loneliness. The first kind of loneliness is temporary.This is the most common type.It usually disappears quickly and does not require any special attention.The second kind is situation，for example，family problem，the death of a loved one，or moving to a new place.Although this kind of loneliness can cause physical problems，such as headaches and sleeplessness, it_usually does not last for more than a year. The third kind of loneliness is the most serious.Unlike the second type，chronic(the opposite of “temporary”)usually lasts more than two years and has no specific cause.People who experience habitual loneliness have problems socializing and becoming close to others.Unfortunately，many chronically lonely people think there is little or nothing they can do to improve their condition. Psychologists(心理学家)agree that one important factor in loneliness is a person's social contacts，for example，friends，family members，co?workers，etc.We depend on various people for different reasons.For instance，our families give us emotional support，our parents and teachers give us guidance，and our friends share similar interests and activities.However，psychologists have found that，though lonely people may have many social contacts，they sometimes feel they should have more.They question their popularity. Psychologists are trying to find ways to help habitually lonely people for two reasons：they are unhappy and unable to socialize and there is a connection between chronic loneliness and serious illness such as heart disease，while temporary and situational loneliness can be a sad，and sometimes dangerous condition. 1.How would you treat temporary loneliness according to the passage? A．Talk to friends. B．Just ignore it. C．Go to see a doctor. D．Ask your teachers for guidance. 2.“It” in the last sentence of the second paragraph refers to ______. A．temporary loneliness B．situational loneliness C．a new place D．sleeplessness 3.Why do psychologists want to help chronically lonely people? A．Chronic loneliness can cause family problems. B．Chronic loneliness can cause serious illness. C．Chronic loneliness can not be overcome. D．Chronic loneliness is incurable. 4.Loneliness is divided according to______. A．the time it lasts B．the places where it appears C．different people it happens to D．different seasons when it appears 5.9What is the best title for the passage? A．Three kinds of loneliness. B．Loneliness and diseases. C．Loneliness and social contacts. D．Chronic loneliness.