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问题 解答题

已知数列a，b，c是各项均为正数的等差数列，公差为d（d＞0），在a，b之间和b，c之间共插入n个实数，使得这n+3个数构成等比数列，其公比为q．

（I）求证：|q|＞1；

（II）若a=1，n=1，求d的值．

答案

（1）由题意知qn+2=

c
a
，c=a+2d，

又a＞0，d＞0，可得qn+2=

c
a
=1+
2d
a
＞1

即|qn+2|＞1，故|q|n+2＞1，又n+2是正数，故|q|＞1．

（2）由a，b，c是首项为1、公差为d的等差数列，故b=1+d，c=1+2d，

若插入的这一个数位于a，b之间，则1+d=q2，1+2d=q3，

消去q可得（1+2d）2=（1+d）3，即d3-d2-d=0，其正根为d=

1+
5
2

若插入的这一个数位于b，c之间，则1+d=q，1+2d=q3，

消去q可得1+2d=（1+d）3，即d3+3d2+d=0，此方程无正根．

故所求公差d=

1+
5
2

单项选择题

可能与龋齿有关的病原菌是

A．白色念珠菌

B．大肠埃希菌

C．变异链球菌

D．艰难梭菌

E．甲型溶血性链球菌
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完形填空

III.完型填空（共15小题；每小题1分，满分15分）

I stayed at home with my 5-year-old daughter Lily. When I was reading some magazines, Lily played in the living room. But after a while I noticed nothing but  31     .           

She had never been so quiet. Was she   32   ? I asked what she was doing. The   33    finally came after I   34   my question three times. I heard “Nothing”. Never could I believe she just stayed there and kept quiet. So I walked into the living room. She hurried off. I   35    her to turn around but she   36  . I got a little angry. “Young lady,” I said, “turn around!”

37   she turned toward me. In her hand was my wife’s new lipstick, only a little   38  . Besides her lips, every inch of her face was covered with bright red.

As she looked up at me with   39   eyes, I really got angry. I was about to   40   at her so that she would know what a mistake she had made.  41   before I could shout, I noticed the big letters she had written on her white skirt with the lipstick, “I’M  A  PERFECT  ANGEL!”

Something strange   42  me. I looked back at her red face. This time, instead of seeing a  43  girl, I saw a little angel that I had almost missed!

“Lily, I heard you speak to the toys just now. I guess you were teaching them how to use the lipstick. But should an angel use her mom’s lipstick like this?” Lily lowered her head, “Sorry. I won’t do it again.” I smiled. “Go and wash your face and let’s get a new  44  for mom.”

Every child is an angel. I felt   45  that I didn’t hurt the angel through treating her mistake in a right way.

31. A. sound          B. laughter       C. silence        D. music

32. A. sleeping        B. laughing       C. running       D. standing

33. A. news           B. result         C. report         D. reply

34. A. read            B. wrote         C. found         D. repeated

35. A. allowed         B. advised       C. ordered        D. persuaded

36. A. accepted        B. forgot         C. refused        D. understood

37. A. Suddenly        B. Slowly       C. Happily        D. Sadly

38. A. left             B. gone        C. lost           D. saved

39. A. excited          B. frightened     C. surprised       D. disappointed

40. A. shout            B. beat         C. drive          D. hold

41. A. But             B. So           C. And          D. Or

42. A. pulled          B. left          C. pushed         D. hit

43. A. foolish          B. bad           C. clever         D. boring

44. A. skirt            B. lipstick        C. present        D. magazine

45. A. sad             B. interesting      C. fortunate      D. sorry
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