# If the AM and GM for 10 observations are both 15 then the value of harmonic mean......?

Jun 8, 2018

15 (Assuming all the observations are nonnegative)

#### Explanation:

The generalized AM-GM theorem says that if ${a}_{1} , {a}_{2} , \ldots , {a}_{n}$ are non-negative, then

$\frac{{a}_{1} + {a}_{2} + \ldots + {a}_{n}}{n} \ge \sqrt[n]{{a}_{1} {a}_{2.} . . {a}_{n}}$

with equality holding only when

${a}_{1} = {a}_{2} = \ldots = {a}_{n}$

Thus, in this case, all the observations (assumed non-negative) must be 15, and their harmonic mean must be the same.