How close can you get by advancing half the distance of your previous step?

Question

If you set on a journey from point A to poin B and each day you advance half the distance that you advanced the previous day, will you ever reach your destination? i.e. Point B is 100 miles from point A. Day 1 I walk 50 miles, day 2 I walk 25 miles, etc. How close can I get while still moving forward? Will I ever stop moving?

Answer 1

This is one of the definitions of infinity. You will eventually reach an infinitesimally small distance from your destination. But you will never reach it even after an infinite number of taking 1/2 the distance.

In calculus, eventually you take the limit to zero, which would be the rate of change of your movement. In your case, this will also approach zero, so you would take the limit of that to zero, which is the rate of acceleration, or the second derivative...and so on.

Answer 2

This is a variant of one of the famous paradoxes attributed to the ancient Greek philosopher/mathematician Zeno.

There are two common answers to this question, a geometric answer and a quantum physics answer. Since this was presented as a math question, I will give the geometric answer first.

GEOMETRY:
In Euclidean geometry, traveling between two geometric points by traveling to the midpoint each day would ensure that you never reach the destination point. Since a geometric point is infintessimally small and since there is always a midpoint between two geometric points is it always possible to travel half the distance without arriving. Hence you will travel forever without arriving.

QUANTUM MECHANICS:
In the real world as understood by quantum mechanics, there will come a time on your journey that you are so close to your destination that the Heisenberg uncertainty principle takes over. You will not be able to tell whether or not you have not arrived or whether or not you can get closer to your destination.

Basically at that time you will be within the statistical margin of error of having arrived at your destination. This is rather like those political polls that say two candidates are 3% apart but that the race is "statistically" a tie because of a 5% margin of error. Hence, at that time, as close as physics can determine, you will have arrived.

Answer 3

If you close only half the distance each day, then no, you'll never arrive, mathematically. For your 100-mile trip, after 10 days, you'd have less than a tenth of a mile to go. After 20 days, you'd have just over 6 inches to go. After 30 days, you'd have less than 0.006 inches left. After 40 days, you'd have less than 0.000006 inches remaining. Of course, over those 10 days, after traveling a whopping distance of 0.005895 inches, most people would treat you as a human statue... even more so after 40 days.

Pre-Arhcimedean thinkers would use this as proof that Achilles could never catch the hare in a race. Pre-Archimedean thinkers today (and believe me, there are WAY too many of them!) try to use this as proof that 0.99999... (repeating forever) is less than one.

Fortunately, if you were traveling at a constant speed, you would indeed make it to your 100-mile mark. Yes, you would pass the 1/2-way mark, the 3/4 mark, the 7/8 mark, and infinitely many others on your way, but by simply dividing your total distance by your speed, you'll know how long it would take to arrive. But with the constraints you've put into this problem, with an ever-slowing speed approaching zero, no, you'll never arrive.

Answer 4

The problem claimed "Each day you adance half the distance that you advanced the previous day."

Whether or not you reach B depends entirely upon how far you step on the first day.

If on the first day you step over half way, then you will indeed reach B in a finite number of steps.

If on the first day, you step less than half way, you will never reach B. In fact, you will always stay some distance away from B.

If on the first day, you step exactly half way, then the limit is B. You never reach B, but you can get arbitrarily close in finitely many steps.

Answer 5

Are you a theorist or do you work with real world solutions? The theorist says that you will never reach your destination. The real world worker, such as a technician or engineer, says that a certain finite distance is close enough. If your toe is .0001 inches from a specific location, then you have arrived.

Answer 6

You will only be able to get about half centimeter without becomming so inaccruate that you can't measure yourself.

But you can have fun all day standing 2 centimeters from your target before moving to 1 centimeter and waiting another day.

I really don't believe you would make it 10 feet from your target before you give up and do something else.

Answer 7

Practically, you will eventually get there, but mathematically, you wont. The sequence you are talking about is given by

a(n) = 50 * 0.5 ^ n

Partial sums are:
a(1) = 50.000 s(1) = 50.000
a(2) = 25.000 s(2) = 75.000
a(3) = 12.500 s(3) = 87.500
a(4) = 06.250 s(4) = 93.750
a(5) = 03.125 s(5) = 96.875
a(6) = 01.563 s(6) = 98.448
a(7) = 00.782 s(7) = 99.230
etc., etc.
You could carry this on forever, but by day 10, you will be practically there.

Answer 8

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Answer 9

Theoreticaly you would never get there. You would get so close that you would only be moving infinitesimal parts of an inch