 # A particle begins at the origin and moves successively, 1 unit to the right, 1/2 unit up, 1/4 unit to the left, 1/8 unit down, 1/16 unit to the right, 1/32 unit up etc.The length of each move is half the length of previous move and movement continues indefinitely.The co-ordinates of the point to which the movement converges is :

9 years ago
x coordinate = 1-(1/4)+(1/16)-(1/64).......
y coordinate = (1/2) -(1/8) + (1/32) .........

converge the above two series to get the X and Y as the result of converging coordinate

Thanks & Regards

Saurabh Singh,

B.Tech.

IIT Kanpur

9 years ago
ans:

total displacement in X direction :
step-1 : 1 unit right (+ve X direction)
step-3 : 1/22unit left (-ve X deirction)
step-5: 1/24unit right ( +ve X direction)
step-7 : 1/26unit ( -ve X direction)
and so on..
so total displacement in X direction: (X-coordinate)
X= 1 - 1/22+ 1/24_1/26+ 1/28 - ........................................ up to infinity
its an G.P. so sum of it :
X = 1/ {1+ 1/22} sum of G.P = a/ (1-r)
1st term = 1 and C.R. = -1/22
X = 4/5

similarly
total displacement in Y direction :
step-2 : 1/2 unit up (+ve Y direction)
step-4 : 1/23unit down (-ve Y deirction)
step-6: 1/25unit up ( +ve Y direction)
step-7 : 1/27unit ( -ve Y direction)
and so on..
so total displacement in Y direction: (Y-coordinate):

Y= 1/2 - 1/23+ 1/25 - 1/27.................................................. up to infinity
its a g.p.
so sum : Y = 1/2*{1 +1/22} 1st term = 1/2 and C.R. = -1/22
Y = 2/5

so coordinates : (4/5 , 2/5)

Thanks and Regards,
Ajay verma,