题目

设数列{an}满足a1=2，a2+a5=14，且对任意n∈N*，函数f（x）=an+1x2﹣（an+2+an）x满足f′（1）=0． （1）求数列{an}的通项公式； （2）设bn=，记数列{bn}的前n项和为Sn，求证Sn＜． 答案：【考点】数列的求和；数列递推式． 【分析】（1）求出函数的导数，由条件可得2an+1=an+2+an，由等差数列的性质可得数列{an}为等差数列，设公差为d，运用等差数列的通项公式，可得d=2，即可得到通项公式； （2）由bn==（﹣），运用裂项相消求和，由不等式的性质，即可得证． 【解答】（1）解：函数f 阅读理解 　　Writing articles about films for The Front Page was my first proper job．Before then I had done bits of reviewing－novels for other newspapers, films for a magazine and anything I was asked to do for the radio．That was how I met Tom Seaton, the first arts editor of The Front Page, who had also written for television．He hired me, but Tom was not primarily a journalist, or he would certainly have been more careful in choosing his staff． 　　At first, his idea was that a team of critics should take care of the art forms that didn’t require specialized knowledge：books, TV, theatre, film and radio．There would be a weekly lunch at which we would make our choices from the artistic material that Tom had decided we should cover, though there would also be guests to make the atmosphere sociable． 　　It all felt like a bit of dream at that time：a new newspaper and I was one of the team．It seemed so unlikely that a paper could be introduced into a crowded market．It seemed just as likely that a millionaire wanted to help me personally, and was pretending to employ me．Such was my lack of self-confidence． 　　Tom’s original scheme for a team of critics for the arts never took off．It was a good idea, but we didn’t get together as planned and so everything was done by phone．It turned out, too, that the general public out there preferred to associate a reviewer with a single subject area, and so I chose film．Without Tom’s initial push, though, we would hardly have come up with the present arrangement, by which I write an extended weekly piece, usually on one film． 　　The space I am given allows me to broaden my argument－or forces me, in an uninteresting week, to make something out of nothing．But what is my role in the public arena? I assume that people choose what films to go to on the basis of the stars, the publicity or the director．So if a film review isn’t really a consumer guide, what is it? I certainly don’t feel I have a responsibility to be ‘right’ about a movie．Nor do I think there should be a certain number of ‘great’ and ‘bad’ films each year．All I have to do is put forward an argument．I’m not a judge, and nor would I want to be． (1) What do we learn about Tom Seaton from the first paragraph? [　　] A． He encouraged the writer to become a writer． B． He met the writer when working for television． C． He had worked in various areas of the media． D． He prefers to employ people that he knows． (2) The weekly lunches were planned in order to ________． [　　] A． help the writers get to know each other B． distribute the work that had to be done C． provide an informal information session D． entertain important visitors from the arts (3) What does the author mean when he says that Tom’s plan never took off in Paragraph 4? [　　] A． Tom’s plan was unpopular． B． Tom’s plan wasted too much time． C． Tom’s plan wasn’t planned properly． D． Tom’s plan wasn’t put into practice． (4) Which of the following best describes what the writer says about his work? [　　] A． He can freely express his opinion． B． He prefers to write about films he likes． C． His success varies from year to year． D． He writes according to accepted rules．