Find the common difference of this A.P.

[utkarshakash]

Homework Statement

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.

Homework Equations

The Attempt at a Solution

$b^2=ac \\ 2log b=log a+log c$
Also
$2\dfrac{log c}{log b}=\dfrac{log a}{log c}+\dfrac{log b}{log a}$

 

[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.

 

[Ray Vickson]

Homework Statement

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.

Homework Equations

The Attempt at a Solution

$b^2=ac \\ 2log b=log a+log c$
Also
$2\dfrac{log c}{log b}=\dfrac{log a}{log c}+\dfrac{log b}{log a}$

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

 

[utkarshakash]

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

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?

 

[Ray Vickson]

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?[/QUOTE

No, they are not useless questions---they are supposed to make you realize that using undefined acronyms is not a good policy. Effective communication is an important part of problem-solving. I am not the only poster who has had problems with your presentations. What, exactly, is so hard about writing out "arithmetic progression" or "geometric progression" (or, for that matter, "harmonic progression")? If you wanted to save space you could say 'arithmetic progression (AP)', then use 'AP' after that in the same posting/thread (but not in different threads).

BTW: I do know some maths, having had honours undergrad work in the subject and a few graduate courses in it at MIT, as well as having taught math-related subjects (Operations Research) at university for more than 30 years. So, that little criticism of yours does not wash.

RGV

 

[haruspex]

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?

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.

 

[utkarshakash]

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.

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

 

[utkarshakash]

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.

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

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

 

[haruspex]

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

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

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.