Answer:
Step-by-step explanation:
(a) In the first step, we remove the middle third of a line segment whose length is 1, which means that the length of the removed middle third is . In the next step, we remove the middle third of the remaining two segments whose lengths are , which means that the length of removed parts in this step is . In the third step we remove middle thirds of four segments which we got in the previous step whose lengths are each, so, length of removed parts is
After the third step, we got eight segments whose lengths are each, which means that we now remove 8 segments whose lengths are , or, their total length is .
The conclusion is that the first four terms of a series that represents the total length of the segments removed after four steps are
(b) According to the previous part, we can conclude that the nth term of this sequence is
and a summation with an nth term that represents the amount removed if this pattern were continued infinitely is
(c) We can notice that the previous sum is actually the geometric progression, where the first term is and the quotient is , so, the required sum is
The conclusion is that this infinite series converges.