A particle P starts at the origin (x0,y0)=(0,0), of the plane. The particle moves in steps as follows.
1. P moves 1 unit to the right to the point (x1,y1)=(1,0), then
2. P moves 1/2 unit up to (x2,y2)=(1,1/2), then
3. P moves 1/4 unit to the left to (x3,y3)=(3/4,1/2), then
4. P moves 1/8 unit down to (x4,y4)=(3/4, 3/8), then
5. P moves 1/16 unit to the right to (x5,y5)= (13/16, 3/8)
6. and so on...
Find the limit point P-infinity= (x-infinity,y-infinity) of the particle P, where the limit to x-infinity= the limit of x sub k . as k approaches infinity and y-infinity= the limit of y sub k, as k approaches infinity.
I have no idea what I'm supposed to do or where to start. Any help will be much appreciated. Q.Q
Take a look at the vertical and horizontal movements separately
vertical: 0 + 1/2 - 1/8 + 1/32 - .... ignoring the 0 , this is a GS where a = 1/2 and r = -1/4
sum (all terms) = a/(1-r) = 1/2 ÷ (5/4) = 2/5, so the y will approach 2/5
horizontal: 0 + 1 - 1/4 + 1/16 - .... again, let a = 1, r = -1/4
again sum of all the terms = a/(1-r) = 1/(5/4) = 4/5
the final position will be ((4/5, 2/5)
Sal Khan explains the formula very nicely here:
https://www.youtube.com/watch?v=b-7kCymoUpg
I am confused as to why the sum of all the terms is a/(1-r). I thought it was (a-ar^n-1)/(1-r).
To find the limit point P-infinity of the particle P, we need to analyze the pattern of the particle's movement.
Let's start by examining the x-coordinate of the particle's position. Notice that at each step, the particle moves to the right, and then to the left, alternating directions. The distance it moves in each step decreases by half compared to the previous step. This forms a geometric series with a common ratio of 1/2.
The x-coordinate of the particle's position after each step is given by the following formula:
xk = x0 + (1 - 1/2 - 1/4 - ... - 1/2^k)
where x0 is the initial x-coordinate (0 in this case), and k is the step number.
If we simplify the sum inside the parentheses, we get:
1 - 1/2 - 1/4 - ... - 1/2^k = 1 - 1/2^k
So, the x-coordinate after k steps becomes:
xk = x0 + (1 - 1/2^k) = 0 + (1 - 1/2^k) = 1 - 1/2^k
Now let's look at the y-coordinate of the particle's position. Similarly, the y-coordinate moves up and down in alternate steps, with the distance decreasing by half each time. It also forms a geometric series with a common ratio of 1/2.
The y-coordinate of the particle's position after each step is given by the following formula:
yk = y0 + (-1/2 + 1/2^2 - 1/2^3 + ... + (-1/2)^(k+1))
Again, if we simplify the sum inside the parentheses, we get:
-1/2 + 1/2^2 - 1/2^3 + ... + (-1/2)^(k+1) = (-1/2)(1 - (-1/2)^k) / (1 - (-1/2))
= (-1/2)(1 - (-1/2)^k) / (1 + 1/2)
= (-1/2)(1 - (-1/2)^k) / (3/2)
= (-2/3)(1 - (-1/2)^k)
So, the y-coordinate after k steps becomes:
yk = y0 + (-2/3)(1 - (-1/2)^k) = 0 + (-2/3)(1 - (-1/2)^k) = (-2/3)(1 - (-1/2)^k)
Now, let's find the limit of xk and yk as k approaches infinity.
lim(k->∞) (1/2^k) = 0 (since the series converges)
Therefore, the limit of xk as k approaches infinity is:
lim(k->∞) (1 - 1/2^k) = 1 - 0 = 1
Similarly, the limit of yk as k approaches infinity is:
lim(k->∞) (-2/3)(1 - (-1/2)^k) = (-2/3)(1 - 0) = -2/3
Hence, the limit point P-infinity of the particle P is (x-infinity, y-infinity) = (1, -2/3).