Maths-
General
Easy

Question

A particle begins at the origin and moves successively in the following manner as shown, 1 unit to the right, 1 halfunit up, 1 fourth unit to the right, 1 over 8 unit down, 1 over 16 unit to the right etc. The length of each move is half the length of the previous move and movement continues in the ‘zigzag’ manner indefinitely. The co-ordinates of the point to which the ‘zigzag’ converges is

  1. (4 over 3, 2 over 3)    
  2. (4 over 3, 2 over 5)    
  3. (3 over 2, 2 over 3)    
  4. (2, 2 over 5)    

hintHint:

formulate the motion of the particle and calculate the infinite sum of the resultant GP

The correct answer is: (4 over 3, 2 over 5)


    ( 4/3, 2/5)
    The x coordinate can be calculated as follows:
    S= 1+ ¼ + 1/16 +….
    Sum of infinite GP is a/(a-r)
    = 1/(1-1/4)
    = 4/3
    Now, in the y direction, the particle forms the following series:
    ½-1/8 +1/32 -….
    Here , we can observe that the value of a = ½ and r = -1/4
    Therefore,
    S= (1/2)/(1-)-1/4))
    =(1/2)/(1+1/4) = (1/2)/(5/4)
    = 2/5
    Coordinates = ( 4/3, 2/5)

    the problem states that the particle’s movement follows a geometric progression with the first term being 1 and the ratio being ½ and the particle moves infinitely.
    The ratio in the x direction is ¼ . The ratio in the y direction is -1/4

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