Question

The arithmetic mean (A) of two numbers (a and b) is given by the formula A=a+b/2, and their geometric mean (G) is given by G=√ab. Their harmonic mean (H) is given by the formula G= √AH. Which formula correctly gives H in terms of a and b?

Answer 1

A = a+b / 2
G = √( a b )
G = √ ( A H )   / ²  ( we will square both sides of the equation )
G² = A H
H = G² / 2
$H = \frac{G ^{2} }{A}= \frac{a b}{A}= \\ = \frac{ab}{ \frac{a+b}{2} }= \frac{2ab}{a+b}$

Answer 2

G = √AH
√ab = √AH
ab = AH
H = ab/A ab / (a + b)/2 = 2ab/(a + b)
H = 2ab/(a + b)