Math 1210-007
Mathematical Proofs
Fall 2010
Geometric Series Proof
Geometric Series: The sum of the first n powers of a number r, which we shall call S n , can be
written concisely in the following way:
n
S n=∑ k=0 r k =1rr 2...r n=
1−r n1
1−r
Proof:
By definition we know that
2
S n=1r r r
n
(1)
Multiplying both sides of the equation by r yields
rS n =r r 2r n1
(2)
Subtracting equation (2) from equation (1) yields the following [Notice all the terms that cancel]
S n−rS n=1−r
n1
Factoring out the quantity S n on the left hand side gives
S n 1−r =1−r n1
Finally, dividing both sides by the quantity 1−r  yields the desired equation
n1
S n=
1−r
1−r
Q.E.D.
Math 1210-007
Mathematical Proofs
Fall 2010
Proof By Induction
Proof by induction is one method of proof for problems where the goal is to demonstrate that a
formula works for all natural numbers n. This is done in two steps.
Step 1: Show that the formula works when n=1
Step 2: Assume that the formula works for a number n and show that it must be the case that the
formula works for n1
To see why this logic works let's try using these two steps on simple examples. Suppose that you have
a million boxes all in a single file straight line and you are standing at the front of that line. Next
suppose I state the following two facts about the boxes:
Fact 1: The box at the front of the line is empty.
Fact 2: If a given box is empty, then the box directly behind that box must also be empty.
Q: What can you conclude from this?
A: You know that the first box is empty. Now by applying “Fact 2” it must be the case that the box
directly behind the first box (i.e. the second box) is empty. Applying “Fact 2” again means that the box
directly behind the second box (i.e. the third box) is empty. If you continue in this way you will see
that “Fact 1” and “Fact 2” together imply that all of the boxes are empty.
Notice that if it was originally your goal to prove that all boxes are empty, then it would have been
sufficient for you to prove “Fact 1” and “Fact 2” are both true. The exact same type of logic works in
mathematical induction, except instead of dealing with boxes you are dealing with natural numbers,
and instead of proving that all boxes are empty you are trying to prove a formula works for all natural
numbers. You first demonstrate that the formula works for the first number ( n=1 ). Then you show
that if the formula works for a given number, n, then it must work for the number directly behind it (i.e.
n1 ). Together these two facts imply that the formula must work for all natural numbers.
Let's consider an example (on the next page) where we use mathematical induction to prove the
Gaussian Sum Formula.
Math 1210-007
Mathematical Proofs
Fall 2010
Gaussian Sum Formula: The sum of the first n natural numbers can be more easily calculated as
follows:
n
∑k =1 k =123n=
nn1
= P n
2
Proof:
Step 1: First we want to show that the formula works when n=1 .
The sum of the first integer is
1
∑k =1 k =1
Using the formula with n=1 gives
P 1=
111 2
= =1
2
2
So the formula works in the case when n=1
Step 2: Assume that the formula works for some number n, then show that it must be the case the the
formula works for n1
n n1
=P  n for some number n. Now lets see what
2
we get when we sum the first n1 integers.
So we are assuming that 123n=
n
∑k =1 k =123n n1=1n n1=
n n1
n1n11
n1=
2
2
The expression on the far right is the result of several steps of algebraic simplification done to the
second to last expression. More importantly, notice that the expression on the far right is equivelant to
P n1 . Therefore, by assuming that the formula works for a number n, we were able to use this
fact to show that the formula must also work for n1 . In other words, we have found that
n1
∑k =1 k =P  n1
.
Let us step back for a second and summarize what we have proven. We have proven that P 1 is
equivalent to the sum of the first integer. We have also proven that if P n is the sum of the first n
integers, then P n1 is the sum of the first n1 integers. Therefore, having proven these two
essential facts, we can apply the logic of mathematical induction to conclude that the Gaussian sum
formula must work for all natural numbers. Q.E.D.