Problem: If a and b are two unequal positive numbers, then:
Answer Choices:
A. a+b2ab>ab>2a+b
B. ab>a+b2ab>2a+b
C. a+b2ab>2a+b>ab
D. 2a+b>a+b2ab>ab
E. 2a+b>ab>a+b2ab
Solution:
The Arithmetic Mean is (a+b)/2, the Geometric Mean is ab, and the Harmonic Mean is 2ab/(a+b). The proper order for decreasing magnitude is ( E );
or
Since (a−b)2>0, we have a2+b2>2ab;∴a2+2ab+b2>4ab, a+b>2ab, and (a+b)/2>ab.
Since a2+2ab+b2>4ab, we have 1>4ab/(a+b)2;∴ab>4a2b2/(a+b)2, and ab>2ab/(a+b).