# Find the common difference of this A.P.

1. Nov 22, 2012

### utkarshakash

1. The problem statement, all variables and given/known data
If a,b,c are distinct positive real numbers in G.P. and $log_c a,log_b c,log_a b$ are in A.P., then find the common difference of this A.P.

2. Relevant equations

3. The attempt at a solution
$b^2=ac \\ 2log b=log a+log c$
Also
$2\dfrac{log c}{log b}=\dfrac{log a}{log c}+\dfrac{log b}{log a}$

2. Nov 22, 2012

### haruspex

I suggest substituting A = log a etc, just to make it easier to read/write.
Since you are asked for the common difference, it might help to invent a variable for that and use it to substitute for some of A, B, C.

3. Nov 22, 2012

### Ray Vickson

Does A.P. mean "Associate Press", "Audio Precision", "Advanced Placement", or what? These are what I get (among many others) when I do Google. To me, G.P. stands for "Geometric Programming", which is a branch of Optimization used widely in engineering and similar fields. Oh.... maybe you mean "arithmetric progression" and "geometric progession", but I cannot say for sure, not being a mind-reader.

RGV

4. Nov 22, 2012

### utkarshakash

If you do know some maths then you MIGHT NOT ask these useless questions. Is common difference related to "Associate Press", "Audio Precision", "Advanced Placement" in any way?

5. Nov 22, 2012

### Ray Vickson

6. Nov 22, 2012

### haruspex

utkarshakash, mind your manners! We're trying to help you and many like you. There are many branches of mathematics. Abbreviations like that are fine once you're tuned into the branch in question, but you must understand that we come into each post cold. On one of your posts today I read, up front, that such and such were in G.P. My first thought was "general position". We're not mind readers. It is up to you to try to make things clear so that we can get the picture easily.

7. Nov 23, 2012

### utkarshakash

OK I won't post questions involving such ambiguity in future.

8. Nov 23, 2012

### utkarshakash

OK as you said I assumed A as log a and so on.

$\dfrac{A}{C},\dfrac{C}{B},\dfrac{B}{A}$ are in Arithmetic Progression.

Also the common difference of this Arithmetic Progression is given by
$\dfrac{\frac{B}{A}-\frac{A}{C}}{3-1}$

To find the value of the numerator I substitute B=(A+C)/2;

After simplifying I get

$\dfrac{1-\frac{C}{A}-\frac{A}{C}}{2}$

9. Nov 23, 2012

### haruspex

What I meant was, let x = C/B, A/C = x-y, B/A = x+y. You should get down to specific values (perhaps several choices) for x and y.