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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?

Respuesta :

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[tex]G=\sqrt{AH} \\ G^2=(\sqrt{AH})^2 \\ G^2=AH \\ H=\frac{G^2}{A} \\ \\ A=\frac{a+b}{2} \hbox{ and } G=\sqrt{ab} \\ \Downarrow \\ H=\frac{G^2}{A}=\frac{(\sqrt{ab})^2}{\frac{a+b}{2}}=\frac{ab}{\frac{a+b}{2}}=ab \times \frac{2}{a+b}=\frac{2ab}{a+b} \\ \\ \boxed{H=\frac{2ab}{a+b}}[/tex]

Answer:

[tex]H = \frac{2ab}{a+b}[/tex]

Step-by-step explanation:

As per the given statement:

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

[tex]A = \frac{a+b}{2}[/tex]                          .....[1]

and

their geometric mean (G) is given by :

[tex]G = \sqrt{ab}[/tex]                             .....[2]

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

[tex]G = \sqrt{AH}[/tex]

Squaring both sides we get;

[tex]G^2 = AH[/tex]

Substitute the given values we have;

[tex](\sqrt{ab})^2 =\frac{a+b}{2} \cdot H[/tex]

⇒[tex]ab = \frac{a+b}{2} \cdot H[/tex]

Multiply by 2 both sides we have;

[tex]2ab = a+b \cdot H[/tex]

Divide both sides by a+b we have;

[tex]\frac{2ab}{a+b} =H[/tex]

or

[tex]H = \frac{2ab}{a+b}[/tex]

Therefore, the formula correctly gives H in terms of a and b is, [tex]H = \frac{2ab}{a+b}[/tex]