Maths problem to solve

21st October 2016, 19:14
(This post was last modified: 21st October 2016, 21:25 by Aractus.)
Okay, so I saw Numberphile use this little example as a throwaway example in one of their videos, so I thought I'd re-frame it as a problem and see if anyone here can solve it.

Imagine you have a pet ant, and he's your favourite pet. You absolutely love him, and you want to keep him entertained while you're away. In the same way that hamsters love running on wheels, your ant loves to walk around the circumference of elastic bands. He thinks it's terrific, but you have to keep buying more elastic bands for him because he never wants to walk the same elastic band twice. At the moment you use elastic bands that are stretched out to a circumference of one metre, and your ant walks around reach one in 100 seconds - it takes him 1 second to move 1cm. You currently need to supply your athletic ant with enough entertainment to last him nine hours a day, which means he goes through 324 huge 1m elastic bands per day!

We'll assume that your ant has been blessed with immortality, as well as infinite stamina. A cleaver entrepreneur has just invented a new type of elastic band that can keep on stretching out forever. You come up with an ingenious idea - if you stretch it out by 1 metre per second it should last your ant a very long time, and you won't have to keep buying hundreds of brand new 1m circumference elastic bands for him every day. In fact you can just buy one and by the time your pet ant finishes, then you can just get another.

So you get the new elastic band that can be stretched forever, it starts with a circumference of 1 metre, and you place your pet ant on it, and he starts walking around the circumference. Every time your ant walks 1cm you will stretch out the elastic band by another metre. So it starts with a circumference of 1 metre, the ant walks 1cm, you stretch it out by a metre, it's now two metres in circumference and the ant walks another centimetre, and then it gets stretched out to three meres, and he walks another centimetre, and this process will keep on going until he completes his journey around the elastic band, just like this:

https://blog.aractus.com/videos/numberphile-ant.mp4

I was going to take a screenshot and show you that, but then I thought I'll just show you a short section of the numberphile video so you can see the elastic band stretch after each 1cm the ant moves.

So the question is this: can your ant ever complete his circuit around the circumference of the elastic band? If not, why not, and if he can how long will it take and how big will the elastic band be at the end?

Assume of course that he never stops walking, 24 hours a day, 7 days a week, until he completes, if he completes his gigantian journey.

(edited to add...) And FYI, if you think you'll just find the Numberphile video and use their answer, the answer they give to this problem is wrong, and I can show you the calculation that proves that. In fact not only is their answer wrong, it's spectacularly wrong. Their answer is not even close to correct. But then again half the stuff they show you like "1 + 2 + 3 + 4 + ... = -1/12" is wrong, but in this instance it's clear they made a miscalculation in their answer to this "throw-away question". If you want a hint as to how to solve this problem, first construct a finite series to describe the problem using maths, and then solve for the value of that series.

Imagine you have a pet ant, and he's your favourite pet. You absolutely love him, and you want to keep him entertained while you're away. In the same way that hamsters love running on wheels, your ant loves to walk around the circumference of elastic bands. He thinks it's terrific, but you have to keep buying more elastic bands for him because he never wants to walk the same elastic band twice. At the moment you use elastic bands that are stretched out to a circumference of one metre, and your ant walks around reach one in 100 seconds - it takes him 1 second to move 1cm. You currently need to supply your athletic ant with enough entertainment to last him nine hours a day, which means he goes through 324 huge 1m elastic bands per day!

We'll assume that your ant has been blessed with immortality, as well as infinite stamina. A cleaver entrepreneur has just invented a new type of elastic band that can keep on stretching out forever. You come up with an ingenious idea - if you stretch it out by 1 metre per second it should last your ant a very long time, and you won't have to keep buying hundreds of brand new 1m circumference elastic bands for him every day. In fact you can just buy one and by the time your pet ant finishes, then you can just get another.

So you get the new elastic band that can be stretched forever, it starts with a circumference of 1 metre, and you place your pet ant on it, and he starts walking around the circumference. Every time your ant walks 1cm you will stretch out the elastic band by another metre. So it starts with a circumference of 1 metre, the ant walks 1cm, you stretch it out by a metre, it's now two metres in circumference and the ant walks another centimetre, and then it gets stretched out to three meres, and he walks another centimetre, and this process will keep on going until he completes his journey around the elastic band, just like this:

https://blog.aractus.com/videos/numberphile-ant.mp4

Video © Numberphile

I was going to take a screenshot and show you that, but then I thought I'll just show you a short section of the numberphile video so you can see the elastic band stretch after each 1cm the ant moves.

So the question is this: can your ant ever complete his circuit around the circumference of the elastic band? If not, why not, and if he can how long will it take and how big will the elastic band be at the end?

Assume of course that he never stops walking, 24 hours a day, 7 days a week, until he completes, if he completes his gigantian journey.

(edited to add...) And FYI, if you think you'll just find the Numberphile video and use their answer, the answer they give to this problem is wrong, and I can show you the calculation that proves that. In fact not only is their answer wrong, it's spectacularly wrong. Their answer is not even close to correct. But then again half the stuff they show you like "1 + 2 + 3 + 4 + ... = -1/12" is wrong, but in this instance it's clear they made a miscalculation in their answer to this "throw-away question". If you want a hint as to how to solve this problem, first construct a finite series to describe the problem using maths, and then solve for the value of that series.