Question

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)

Answer 1

Answer: just delete my answer

Explanation:

Answer 2

a.


`\displaystyle\frac1a-\frac1h=\frac1h-\frac1b`

`\implies\displaystyle(h-a)/(ah)=(b-h)/(bh)`

`\implies\displaystyle(h-a)/(b-h)=(ah)/(bh)`

`\implies\displaystyle(h-a)/(b-h)=\frac ab`

b.


`\displaystyle h=(2ab)/(a+b)`

`\displaystyle\implies(h-a)/(b-h)=((2ab)/(a+b)-a)/(b-(2ab)/(a+b))`

`\displaystyle\implies(h-a)/(b-h)=(2ab-a(a+b))/(b(a+b)-2ab)`

`\displaystyle\implies(h-a)/(b-h)=(ab-a^2)/(b^2-ab)`

`\displaystyle\implies(h-a)/(b-h)=(a(b-a))/(b(b-a))`

`\displaystyle\implies(h-a)/(b-h)=\frac ab`

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