2015 · Q42
The geometric mean of three positive numbers a, b, c is 3 and the geometric mean of another three positive numbers d, e, f is 4. Also, at least three elements in the set {a, b, c, d, e, f} are distinct. Which one of the following inequalities gives the best information about M, the arithmetic mean of the six numbers?
Elementary MathematicsStatistics
- AM>2{3}
- BM > 3.5
- CM ≥ 3.5
- DIt is not possible to set any precise lower limit for M
Explanation
By AM ≥ GM, a+b+c ≥ 3(abc)^{1/3} = 9 and d+e+f ≥ 3(def)^{1/3} = 12, so the six-number mean M ≥ 21/6 = 3.5, with equality only for a=b=c=3, d=e=f=4. That case has only two distinct values, but at least three elements are distinct, so M > 3.5 strictly (option B).