Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

If H is the harmonic mean between $a$ and $b$,then show that $$\frac{1}{H-a}+\frac{1}{H-b} = \frac{1}{a} + \frac{1}{b}$$ and $$\frac{H+a}{H-a}+\frac{H+b}{H-b} = 2$$

I substituted $\displaystyle H = \frac{2ab}{a+b}$, then tried some algebraic manipulation, but I am not getting there.

Are they even valid? If yes, could somebody give me some ideas how to approach these?

share|cite|improve this question
Your method should work. If you show your work here, we can point your error. Other possibilites are: Notice that the right-hand-side of the first equation is $2/H$. Multiply by all three denominators, subtract $2H^2$ and solve for $H$. – Phira May 13 '11 at 17:07
If you are worried about validity, you can just try some values. I put it into Excel and they appear to be. It's not a proof, but it can be a disproof. – Ross Millikan May 13 '11 at 17:14
They are not valid if $a=b$. – Aryabhata May 13 '11 at 17:23
Note what @Aryabhatta says. Whatever he says will always be useful. – user9413 May 13 '11 at 17:37
up vote 5 down vote accepted

I don't know why you aren't getting this. You have \begin{align*} \frac{1}{H-a} + \frac{1}{H-b} &= \frac{1}{\frac{2ab}{a+b} -a} + \frac{1}{\frac{2ab}{a+b}-b} \\ &= \frac{a+b}{ab-a^{2}} + \frac{a+b}{ab-b^{2}} \\ &= (a+b) \cdot \biggl[ \frac{1}{a(b-a)} + \frac{1}{b(a-b)}\biggr] \\ \end{align*}

I think this should help you out. Try taking the factor $(a-b)$ common. For the next problem also again substitute the value of $H$ and try doing manipulations. Be patient, and be careful, with your calculations, you shall arrive at the result.

share|cite|improve this answer
I have tried this far before,but I don't understand what exactly meant by taking the factor $(a-b)$ common? – Max May 13 '11 at 18:56
1:Just i used a pencil and paper ... I do get it from there thanks. – Max May 13 '11 at 19:03

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.