4

4.

Write an expression for the sum of the intervals of time traveled by the tortoise:

5.

Some mathematical notation can be helpful here: 2nmeans multiply 2 times itself
ntimes. So,

,

,

. What is

25?

Also, an arithmetic fact is that 1
22

=1?

2

?

?
?

2
. Check it; both the left-hand and right-

hand side of the equation equal 1/4. Similarly,

. Check it. What is the

value?

6.

Combining what we did earlier with the results in #5, do you agree that we can

rewrite our earlier statements as:

Sum of distances traveled by Achilles:

=1

+1?

2

?

?
?

+1?

2

?

?
?

2
+1?

2

?

?
?

3

+L

Sum of distances traveled by Tortoise:

=1? 2

?

?
? +1?

2

?

?
?

2
+1?

2

?

?
?

3

+L

7.

The above sums (that are unending and go on and on) are both examples of what
is called a geometric series. They are not at all sums in the usual arithmetic
meaning of that word. If the sum stops after a finite number of terms, we call it a
finite geometric sum. We can add up the terms of a finite geometric sum using the
usual rules of arithmetic.

Replacing 1/2 by r, we can develop a general formula for the sum of a finite
geometric sum for any rstrictly between 0 and 1 as follows: (Do not be
concerned if you don't remember your algebra well enough to work out the details
establishing the first equation. We will be working with the second formula
below. Look at the second formula and see if you recognize what the symbols
mean and how to use it. For example, write out the formula when n=3and
r=1 / 2. You can get a start on this by looking at the bracketed comment