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Igoryamba
2 years ago
15

Harmonic mean if a number h such that (h-a)/(b-h)=a/b Prove h is H(a,b) iff satisfies either relation a. (1/a)-(1/h) = (1/h) - (

1/b) b. h = (2ab)/(a+b)
Mathematics
1 answer:
Fudgin [204]2 years ago
7 0
A.

\displaystyle\frac1a-\frac1h=\frac1h-\frac1b
\implies\displaystyle\frac{h-a}{ah}=\frac{b-h}{bh}
\implies\displaystyle\frac{h-a}{b-h}=\frac{ah}{bh}
\implies\displaystyle\frac{h-a}{b-h}=\frac ab

b.

\displaystyle h=\frac{2ab}{a+b}
\displaystyle\implies\frac{h-a}{b-h}=\frac{\frac{2ab}{a+b}-a}{b-\frac{2ab}{a+b}}
\displaystyle\implies\frac{h-a}{b-h}=\frac{2ab-a(a+b)}{b(a+b)-2ab}
\displaystyle\implies\frac{h-a}{b-h}=\frac{ab-a^2}{b^2-ab}
\displaystyle\implies\frac{h-a}{b-h}=\frac{a(b-a)}{b(b-a)}
\displaystyle\implies\frac{h-a}{b-h}=\frac ab

The other direction can be proved by following the manipulations in the reverse order.
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