Quant Exotica Use of Arithmetic and Harmonic Progressions in Time, Speed, and Distance  
Here is a textbook situation in Time, Speed and Distance: A man goes from point A to point B with velocity v_{1} and returns with velocity v_{2}. What is his average velocity? Using the formula , the average velocity can be found to be . So far so good. Ever wondered why we get this result? Why do we get the velocity as the harmonic mean of the two velocities? The answer lies in basics of arithmetic and harmonic progressions. Let t_{1} and t_{2} be the time taken while going from A to B and coming back. The situation is shown below: Now what will a hypothetical average velocity in this situation mean? It will mean that a person takes the same time, t_{average}, while going from A to B and coming back. The situation is summarized below: The total time taken will be same as t_{1} + t_{2} > 2 Ã— t_{average} = t_{1} + t_{2} or t_{average} = (t_{1} + t_{2})/2 In another words, t_{1}, t_{average}, and t_{2} will be in arithmetic progression. So how is this related to velocity? Remember that when distance is constant velocity is inversely proportional to time? I.e. V is proportional to 1/T or V = k/T. If T_{1}, T_{2}, and T_{3} were in arithmetic progression, then 1/T_{1}, 1/T_{2}, 1/T_{3} are in harmonic progression => k/T_{1}, k/T_{2}, k/T_{3} are in harmonic progression => V_{1}, V_{2} and V_{3 }are in harmonic progression! It can also be proved that if V_{1}, V_{2} and V_{3 }are in arithmetic progression, then T_{1}, T_{2}, and T_{3} are in harmonic progression. So hereâ€™s the rule:
Now letâ€™s apply these rules in practical problems:
I shall have to end here and leave the rest of it for my CBT Club students. I shall cover some problems based on this in the CBT Club this week. You might also like: Time, Speed and Distance Arithmetic Progression, Geometric Progression and Miscellaneous 
Re: Quant Exotica Use of Arithmetic and Harmonic Progressions in Time, Speed, and Distance  
Re: Quant Exotica Use of Arithmetic and Harmonic Progressions in Time, Speed, and Distance  
Great TG 
Re: Quant Exotica Use of Arithmetic and Harmonic Progressions in Time, Speed, and Distance  
Ans to 5th part is 40 min.. Plz tell if it is correct.. 
Re: Quant Exotica Use of Arithmetic and Harmonic Progressions in Time, Speed, and Distance  
gr8 article.....thnx a lot TG.... 
Re: Quant Exotica Use of Arithmetic and Harmonic Progressions in Time, Speed, and Distance  
thanx a ton tg ur simply great!!!!!!!!!!!! plz give some more probs for practice 
Re: Quant Exotica Use of Arithmetic and Harmonic Progressions in Time, Speed, and Distance  
thanq so much for description of these solutions.thanq so much 5th problems ans is 40 tell me is it right 
Re: Quant Exotica Use of Arithmetic and Harmonic Progressions in Time, Speed, and Distance  
Simply Awesome , you are the Best Donkey after the one in Shrek 
Re: Quant Exotica Use of Arithmetic and Harmonic Progressions in Time, Speed, and Distance  
i think the answer to 5th question will be 45 
Re: Quant Exotica Use of Arithmetic and Harmonic Progressions in Time, Speed, and Distance  
no.. its 40........ harmonic mean of 30 and 60 ... 
Re: Quant Exotica Use of Arithmetic and Harmonic Progressions in Time, Speed, and Distance  
Gr8 Article TG.. U Rock!!!! 
Re: Quant Exotica Use of Arithmetic and Harmonic Progressions in Time, Speed, and Distance  
Hey Tg, After this article things look easy.... Thanx!!!

Re: Quant Exotica Use of Arithmetic and Harmonic Progressions in Time, Speed, and Distance  
The ans. to the last problem would be 40 i.e. the HM of 30 ans 60.... Temme TG, Is that correct???? 
Re: Quant Exotica Use of Arithmetic and Harmonic Progressions in Time, Speed, and Distance  
hi TG U r aricle is awesome.
