Question

The arithmetic mean (A) of two numbers (a and b) is given by the formula A=a b2, 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

`G=√(AH) \\ G^2=(√(AH))^2 \\ G^2=AH \\ H=(G^2)/(A) \\ \\ A=(a+b)/(2) \hbox{ and } G=√(ab) \\ \Downarrow \\ H=(G^2)/(A)=((√(ab))^2)/((a+b)/(2))=(ab)/((a+b)/(2))=ab * (2)/(a+b)=(2ab)/(a+b) \\ \\ \boxed{H=(2ab)/(a+b)}`

Answer 2

`H = (2ab)/(a+b)`

Explanation:

As per the given statement:

The arithmetic mean (A) of two numbers (a and b) is given by the formula:

`A = (a+b)/(2)` .....[1]

and

their geometric mean (G) is given by :

`G = √(ab)` .....[2]

Their harmonic mean (H) is given by the formula:

`G = √(AH)`

Squaring both sides we get;

`G^2 = AH`

Substitute the given values we have;

`(√(ab))^2 =(a+b)/(2) \cdot H`

⇒
`ab = (a+b)/(2) \cdot H`

Multiply by 2 both sides we have;

`2ab = a+b \cdot H`

Divide both sides by a+b we have;

`(2ab)/(a+b) =H`

or

`H = (2ab)/(a+b)`

Therefore, the formula correctly gives H in terms of a and b is,
`H = (2ab)/(a+b)`