62Clarification!
Home at x=0
Office at x=1
He goes from Home to office. Midway is at (0+1)/2 = 1/2
Then returns to the Home . Mid way is (1/2+0)/2 = 1/4
Then returns to Office. Midway is (1/4+1)/2 = 5/8
Then returns to Home. Midway is (5/8+0)/2 = 5/16
Again To Office Midway will be (5/16+1)/2 = 21/32..
SO on and so forth!
62
Lokesh Verma
·2008-11-12 04:59:10
no! that is the thing it seems like.. isnt it :)
but just imagine if he is at 1/2 and the next position is towards the office, he will reach (1/2+1)/2 =3/4
and if the next is towards home, he will reach (1/2+0)/2 = 1/4
so definitely it will not be a limiting case?
9Edited: By Nishant to hide the Answer!
62
Lokesh Verma
·2008-11-12 05:16:06
Yes good work Celestine :)
This is a very Famous question called the Vacillating Mathematician's problem. The interesting thing about the limit here is that it is not a single point. The first conclusion seems to be that he will reach the center! but that, as u can see is not the right solution!
Btw i have edited ur post to hide the correct solution!
9
Celestine preetham
·2008-11-12 05:21:04
dear nishant
when u edit to hide an ans
it becomes visible when one selects part using cursor
62
Lokesh Verma
·2008-11-12 05:22:56
Hahah.
yes Celestine .. that is the basic idea..
so that it becomes visible to those who want it to be visible! :)
One can use the hide link when u want to give hints etc :)
1
taking home towards office as positive and backwards as negative...
wat the man actaully does is...
his displacement : +1/2 - 1/4 + 3/8 - 5/16 + 11/32 - ......
== (2-1)/2 - (21-20)/22 + (22-(21-20))/23 +.....
general term is : (-1)k-1[2k-1 - 2k-2 + .... +2k-(k-1) - 2k-k]/2k
= (-1)k-12k[1/2 -1/4 +1/8 - 1/16 +..... +1/2k-1 + 1/2k]/2k
=(-1)k-1[1/2((-1/2)k-1)/(-1/2-1)]
for k->infinity, the above expression tends to (-1^(k-1))/3.....
so position of the man afetr infinity time,,, x=1/3 or 2/3.
correct me, if wrong.......
62
Lokesh Verma
·2008-11-12 06:48:25
Yes sky.. wonderful solution :)
Good work..
Now what will be his position at t tending to infinity!
I mean what will happen when it tends to infinity ?
62
Lokesh Verma
·2008-11-12 07:23:15
There is slightly more to it indraneel :)
2/3 and a bit more.. and then a slight explanation of it
but it is good that u reached so far :) well done
1yes i think i got it it will be 1/3 or 2/3 .
62
Lokesh Verma
·2008-11-12 11:36:38
So sky's solution above gives clearly that there are two points one at 1/3 and another at 2/3
So we need to understand what happens.
he basically reaches 1/3 then moves back towards office. Then returns from 2/3 which is midway between 1/3 and 1. Again he moves back from 2/3 to 0. the midway of this is 1/3.
So he keeps oscillating (Vacillating) between these two points. Hence this problem get its name "Vacillating Mathematician"
It is a special case of limits which shows that simple looking situations may not lead to a point limit. but to two point limits.
According to the definition of limits that we know, It will give "Limit does not exist". But just to give a feeel. this is what is happening here. I hope u guys could get some feel good by knowing this problem. I know this wont help u directly in IIT. But that this inspires u to like mathematics! :)