Recall Xeno\'s paradox: an arrow is fired at a wall (which we say is 160 m away)
ID: 2878102 • Letter: R
Question
Recall Xeno's paradox: an arrow is fired at a wall (which we say is 160 m away). We take snapshots of the arrow when it's travelled half of the distance, then when it's travelled half of the remaining distance, and so on. Xeno says that, since this process continues forever, the arrow will never hit the wall. Explain what Xeno's paradox has to do with the maths we've seen in Real Analysis. Discuss both the distance of the arrow from the wall and the total distance travelled by the arrow, using sequences and series. Mathematically, how do we resolve Xeno's paradox? Don't write too much: you should be able to explain this in around 150 words, or less than a side of handwriting on A4 paper.Explanation / Answer
an initial distance of say 160 m, we have,
t = 1 + 1 / 2 + 1 / 22 + 1 / 23 + ....... + 1 / 2n
Difference = 160 /2n m
Now we want to take the limit as n goes to infinity to find out when the distance between the body in
apparent motion and its said target at is zero. If we define
Sn= 1 + 1 / 2 + 1 / 22 + 1 / 23 + ....... + 1 / 2n
then, divide by 2 and subtract the two expressions:
S n - 1/2 S n = 1 - 1 / 2n+1
or equivalently, solve for Sn:
Sn = 2 ( 1 - 1 / 2n+1)
So that now S n is a simple sequence, for which we know how to take limits. From the last expression it
is clear that:
lim Sn = 2
as n approaches infinity.
Therefore, Zeno's infinitely many subdivisions of any distance to be traversed can be mathematically
reassembled to give the desired finite answer.
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