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

A. H= 2ab/a+b

B. H= square root of 2ab/a+b

C. H= a+b/2ab

D. H= ab/2(a+b)

Question

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

A. H= 2ab/a+b

B. H= square root of 2ab/a+b

C. H= a+b/2ab

D. H= ab/2(a+b)

Mercury · Answer

$$A = \frac{ a+b }{ 2 }$$
$$G = \sqrt{ab}$$
$$G = \sqrt{AH}$$

since you have two formulas for G, you can set sqrt(ab) and sqrt(AH) equal to each other

$$\sqrt{ab}= \sqrt{AH}$$

you can square both sides to get rid of the square root

ab = AH

now, you also know capital A in terms of lowercase a and b. plug in $A = \frac{ a+b }{ 2 }$ into the equation, and solve for H.