Series Question Involving Logarithms (1 Viewer)

[Amleops]

If the positive numbers a, b, c are three consecutive terms in a geometric sequence, show that log(base10)a, log(base10)b, log(base10)c are three consecutive terms in an arithmetic sequence.

I've fiddled around a lot and I can't seem to get anything worthwhile. Any help would be greatly appreciated.

 

[deswa1]

<a href="http://www.codecogs.com/eqnedit.php?latex=\textup{If it is an A.P., then logc-logb=logb-loga}\\ logc-logb=log(\frac{c}{b})\\ logb-loga=log\frac{b}{a}\\ a,b,c\textup{ are in a geometric progression so }\frac{c}{b}=\frac{b}{a} \\\therefore logc-logb=logb-loga \textup{ and it is an arithmetic sequence}" target="_blank"><img src="http://latex.codecogs.com/gif.latex?\textup{If it is an A.P., then logc-logb=logb-loga}\\ logc-logb=log(\frac{c}{b})\\ logb-loga=log\frac{b}{a}\\ a,b,c\textup{ are in a geometric progression so }\frac{c}{b}=\frac{b}{a} \\\therefore logc-logb=logb-loga \textup{ and it is an arithmetic sequence}" title="\textup{If it is an A.P., then logc-logb=logb-loga}\\ logc-logb=log(\frac{c}{b})\\ logb-loga=log\frac{b}{a}\\ a,b,c\textup{ are in a geometric progression so }\frac{c}{b}=\frac{b}{a} \\\therefore logc-logb=logb-loga \textup{ and it is an arithmetic sequence}" /></a>

 

[bleakarcher]

Let the common ratio in the geometric series {a,b,c} be r.
=>b=ar (1) c=br (2)
Assume {log[10](a), log[10](b), log[10](c)} are in a geometric sequence with common difference d.
=>d=log[10](b)-log[10](a)=log[10](c)-log[10](b)
i.e. log[10](b/a)=log[10](c/b)
From (1): r=b/a
From (2): c/b=r
:.c/b=b/a
Hence, it follows that {log[10](a), log[10](b), log[10](c)} is a geometric sequence.

 

[Amleops]

Ah, I over-complicated it too much......

Thanks a lot

 

[soloooooo]

I don't know the answer to this (could probably find it if really necessary though), although once you enter the real world, this sort of mathematics is useless (unless you were to go into Academia).

 

[bleakarcher]

I don't know the answer to this (could probably find it if really necessary though), although once you enter the real world, this sort of mathematics is useless (unless you were to go into Academia).

randomness ftw?

 

[soloooooo]

randomness ftw?

No, I just don't see the point of questions like this (question OP referred to). Once you finish high school you will never use that stuff again (certainly with regards to proving log consecutive terms). I fail to see why the syllabus would even require any questions even remotely like that.

In the real world, the people that are smart at mathematics will eventually go on to become project managers and into management roles (if they don't want to be stuck in lower paid jobs) where they will use basic (year 8?) mathematics only.

 

[Amleops]

No, I just don't see the point of questions like this (question OP referred to). Once you finish high school you will never use that stuff again (certainly with regards to proving log consecutive terms). I fail to see why the syllabus would even require any questions even remotely like that.

It's a maths assignment, my teacher set it. Blame her

 

[AAEldar]

No, I just don't see the point of questions like this (question OP referred to). Once you finish high school you will never use that stuff again (certainly with regards to proving log consecutive terms). I fail to see why the syllabus would even require any questions even remotely like that.

In the real world, the people that are smart at mathematics will eventually go on to become project managers and into management roles (if they don't want to be stuck in lower paid jobs) where they will use basic (year 8?) mathematics only.

Critical thinking is a big part of Mathematics. Granted I am biased in the fact that it's what I'm studying but I agree that maybe this in particular won't be used every again after school, but the development and understanding that you go through is a good thing.

However I don't think you quite appreciate just how much maths is used in the real world.

 

[bleakarcher]

The creation of the computer/phone what ever the hell your using right now relied heavily on advanced maths. Without it, it wouldnt be there.

 

[soloooooo]

The creation of the computer/phone what ever the hell your using right now relied heavily on advanced maths. Without it, it wouldnt be there.

I understand that, although people of that calibre are rare. The reality is that most people (even with uni qualifications) will have no impact on society.

 

[bleakarcher]

^ Appreciate the study itself. That's not the point.

 

[Carrotsticks]

Solooo, please allow me to reinforce a few points, which will perhaps get you thinking a bit more:

1. Yes, 99% of the Mathematics we learn in the HSC will not be used in real life (it has MANY MANY applications, but we will never PERSONALLY use it) except for perhaps the Financial component of 2U (which is usually done by computers nowadays). Yes, I will never personally apply knowledge of Metric Spaces or Hamiltonian Dynamics to my personal life, but the study of it and the skills involved enrich me overall.

2. As bleakarcher wisely said, everything in the world surrounding us including the computer you are using right now... is not possible without Mathematics.

3. As AAEldar wisely said, it is the THINKING skills that we adapt. For example when first I learnt how to solve the Rubik's cube, I applied much of the thinking skills from it to Mathematics. No, I never needed any Group Theory (the mathematics behind the cube), but I APPLIED what I learnt from it, and it overall helped me as a Mathematics student. Similarly, we employ the critical thinking skills and methods from Mathematics to real life to help us become more successful overall. You will find that in the work place, there are many problems. There are infinite solutions to these problems, but of course we want the optimal one. We employ the problem solving and thinking skills of Mathematics to do this optimally.

4. I find it insulting when you assert that people of a Mathematics background have "no" impact on society. Imagine a world full of people with no problem solving skills. Where would society be?