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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?

Verified Answer :

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

Answer:

\(H = \frac{2ab}{a+b}\)

Step-by-step explanation:

As per the given statement:

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

\(A = \frac{a+b}{2}\)                          .....[1]

and

their geometric mean (G) is given by :

\(G = \sqrt{ab}\)                             .....[2]

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

\(G = \sqrt{AH}\)

Squaring both sides we get;

\(G^2 = AH\)

Substitute the given values we have;

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

⇒\(ab = \frac{a+b}{2} \cdot H\)

Multiply by 2 both sides we have;

\(2ab = a+b \cdot H\)

Divide both sides by a+b we have;

\(\frac{2ab}{a+b} =H\)

or

\(H = \frac{2ab}{a+b}\)

Therefore, the formula correctly gives H in terms of a and b is, \(H = \frac{2ab}{a+b}\)