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Mathematics 5
Winter Term 2000
The World According to Mathematics

Dwight Lahr and Josh Laison

Wednesday Discussion: Week #6

Part 1: Geometric Sums

Consider for a positive number rthe geometric sum:

Sn

=1

+r+r

2

+r

3

+r

4

+r

5+L+r

n

Exercise 1: What is the value of the above sum for r

= 1
2

and n=6?

Exercise 2: By multiplying both sides above by (1-r), verify that:

Sn(1-r)

=

1

+r+r

2

+r

3

+r

4

+r

5+L+r

n

1

-r

=1

-rn+1

[Here are the steps in the multiplication of the right-hand-side by (1-r):

1

+r+r

2

+r

3

+r

4

+r

5+L+r

n

1-r

=
=

1

+r+r2+r3+r4+r5+L+rn
-rn+1

-

r+r

2

+r

3

+r

4

+r

5+L+rn+1

1

Is this what you got when you did it?]

Now, dividing through by 1

-rwe get that:

Sn=1

+r+r

2

+r

3

+r

4

+r

5+L+r

n

1-rn+1
=
1-r

You should find this formula to be useful when you analyze the formulations of Zeno's
paradox that we will study next week.

Also, think about what it might mean if, starting with the above formula, we let n→∞
(that is, we let nget large without bound). In mathematical terms, we describe this
process by writing the right-hand side as:

lim
n→ ∞

1

-rn+1
1-r