题目

已知数列满足，且． Ⅰ使用数学归纳法证明：； Ⅱ证明：； Ⅲ设数列的前n项和为，证明：． 答案：（I）详见解析；（II）详见解析；（III）详见解析. 【解析】 Ⅰ利用数学归纳法，分别讨论当时和当时的情况； Ⅱ，由Ⅰ知，故； Ⅲ因为，所以，由可得，进而可表示出，利用裂项相消法即可求出，从而证得． 【详解】 解：Ⅰ当时，，故当时命题成立； 假设时命题成立，即， 当时，注意在单调递增 完形填空      Not long ago I wasn't doing very well in exams and I was full of disappointment and  __1__ thinking. While on the Internet, I met many new friends, but one in particular __2__. His nickname is Colorful Day and we __3__ on the Web. He told me: he lost his __4__ at age 15, and felt the whole world had closed its door to him.      He was __5__ so fast that I couldn't believe he was blind! But he continued to __16__: Five years ago, I realized it was time to change my life. I had read about Helen Keller and understood her life and __7__. So I developed a website about her life __8__ mine called Three Days to See.      I visited his website which was well produced and vividly__9__.      I asked, "How did you __10__ to build such a website?" He said, "Owing to the new technology, I can use software specially __11__ for the blind. __12__ I point the cursor(光标), it will just read it out to me. We can do far more things and __13__ more information than we could before."      I then asked, "Didn't you have great difficulty with it?"      He hesitated and then began his __24__ with a smiling face, "Yes, but I have faith in myself to __15__ my  colorful day. And modern technology gives me this __16__ to do it."      His __17__ attitude towards life made me believe that our destiny offers not the cup of despair,  __18__ the chalice (大杯) of opportunity. I returned to my lessons, and this time with a smile on my face and with __19__ of a colorful future.      The Internet, which opened to me to __20__ different lives and cultures, has a charm you couldn't find in another wise nonvirtual world. (     )1. A. positive     (     )2. A. turned out   (     )3. A. argued       (     )4. A. eyesight     (     )5. A. speaking     (     )6. A. follow       (     )7. A. friendship   (     )8. A. as well as   (     )9. A. offered      (     )10. A. try         (     )11. A. designed    (     )12. A. Whether     (     )13. A. send        (     )14. A. webpage     (     )15. A. create      (     )16. A. moment      (     )17. A. hopeful     (     )18. A. but         (     )19. A. eagerness   (     )20. A. touch       B. active        B. stood out     B. interviewed   B. hearing       B. saying        B. write         B. marriage      B. such as       B. presented     B. come          B. distinguished B. However       B. access        B. speech        B. produce       B. privilege     B. disappointed  B. and           B. expectations  B. feel           C. negative     C. came out     C. told         C. courage      C. typing       C. read         C. happiness    C. like         C. delivered    C. work         C. planned      C. Whatever     C. communicate  C. words        C. recognize    C. opportunity  C. pleasant     C. so           C. richness     C. experience   D. instructive   D. looked out    D. chatted       D. confident     D. spelling      D. copy          D. hardships     D. as            D. found         D. manage        D. imagined      D. Wherever      D. exchange      D. opinions      D. organize      D. occasion      D. optimistic    D. then          D. nervousness   D. suffer