Question
A particle begins at the origin and moves successively in the following manner as shown, 1 unit to the right, 
unit up, 
unit to the right, 
unit down, 
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
- (
, 
)
- (,
)
- (
, )
- (2, )
, )
, )
, )
)
Hint:
formulate the motion of the particle and calculate the infinite sum of the resultant GP
The correct answer is: (, )
( 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
Related Questions to study
If a1, a2, a3, ........., an are positive real numbers whose product is a fixed number c, then the minimum value of a1 + a2 + a3 + .... + an – 1 + 2an is
the Am- Gm relation is a handy tool for solving sequence related problems. it is applicable to any sequence of numbers.
If a1, a2, a3, ........., an are positive real numbers whose product is a fixed number c, then the minimum value of a1 + a2 + a3 + .... + an – 1 + 2an is
the Am- Gm relation is a handy tool for solving sequence related problems. it is applicable to any sequence of numbers.
Let 
and 
are two arithmetic sequences such that 
and 
then the value of 
is
an A.P is a sequence of mathematical terms which have a common difference with their adjacent elements
Let and are two arithmetic sequences such that and then the value of is
an A.P is a sequence of mathematical terms which have a common difference with their adjacent elements
If a, b and c are three consecutive positive terms of a G.P. then the graph of y = ax2 + bx + c is
Nature of graph of a quadratic equation is given by its discriminant.
Here, a = a, b= ar, c = ar2 , where r is the common ratio of the GP
Given, a,b,c>0
If a, b and c are three consecutive positive terms of a G.P. then the graph of y = ax2 + bx + c is
Nature of graph of a quadratic equation is given by its discriminant.
Here, a = a, b= ar, c = ar2 , where r is the common ratio of the GP
Given, a,b,c>0
If ln (a + c) , ln (c – a), ln (a – 2b + c) are in A.P., then :
an A.P is a sequence of mathematical terms which have a common difference with their adjacent elements
An = A1+(n-1)x d
A2= A1 + d
A3 = A1 + 2d
A3 = A2+d
A3-A2=A2-A1
If ln (a + c) , ln (c – a), ln (a – 2b + c) are in A.P., then :
an A.P is a sequence of mathematical terms which have a common difference with their adjacent elements
An = A1+(n-1)x d
A2= A1 + d
A3 = A1 + 2d
A3 = A2+d
A3-A2=A2-A1
