1、Mathematical Database Page 1 of 15 INEQUALITIES UNIT 1 CLASSICAL INEQUALITIES 1. Inequality of the Means To motivate our discussion, lets look at several situations. (A) A man drove for 2 hours. In the first hour he travelled 16 km, and in the second hour he travelled 32 km. What is his average spee
2、d for the whole journey? (B) A man drove from City P to City Q at a speed of 16 km/h and returned at 32 km/h. What is his average speed for the whole journey? (C) In the traditional Chinese game mahjong, the value (in order not to encourage gambling we use the term value in place of payoff) of a han
3、d is determined by the number of points of the hand and usually grows in a geometric sequence. For instance, Points 0 1 2 3 4 5 6 Value 4 8 16 32 64 128 256 But in this case the value grows too fast. Some people want to modify it so that the value doubles every 2 points rather than every point. For
4、instance, Points 0 1 2 3 4 5 6 Value 4 x 8 y 16 z 32 What should be x, y and z? Some people tend to simply put 6 x =, 12 y = and 24 z = , but then the numbers in the second row will no longer form a geometric sequence. Can we choose x, y, z so that 4, x, 8, y, 16, z, 32 form a geometric sequence? These questions should be fairly easy. For (A), the average speed is 16 32 24 km/h 2 + = .
