Notes taken by Lizzy Elrod (8/9/06)
Proof Techniques:
Proving if, then statements
Direct Proof:
1. assume the ‘if’ part
2. make logical deductions
3. arrive at the ‘then’ part
ex 1: If a polygon has 3 sides, then the sum of its degrees is 180.
The angle marked A in the triangle is equal to the other angle marked A because they
are alternate interior angles formed by a transversal cutting two parallel lines.
Similarly, the angle marked B in the triangle is equal to the other angle marked B for
the same reason. Thus, it can be seen that the sum of the angles of a triangle are the
same as the number of degrees in a straight angle, which is 180, since 360 degrees is
a full revolution.
ex 2: x 2  6 x  8  0
( x  2)( x  4)  0
x  2  0, x  4  0
x  2orx  4
The factoring part is a logical deduction which follows from the previous step. Next,
if two values multiply to zero, at least one of them has to be zero. It them follows that
one of the two terms is zero. Finally, we can add 2 to both sides of the first equation
and 4 to both sides of the second to determine the two possible values for x. Each of
these steps was a logical deduction that followed from previous steps and known
rules, thus, this is a direct proof.
n(n  1)
, using a
2
direct proof method. We also derive the sum of a geometric sequence of n terms with
the first term 1.
ex 3: Here we prove that the sum of the first n positive integers is
n
n 1
S  i
S   xi
i 1
i 0
n
 S   (n  1  i)
i 1
n 1
xS  x  xi
i 0
n
2S   i  (n  1  i)
i 1
n
2S   (n  1)
i 1
n
n 1
xS   xi 1   xi
n
i 0
i 1
n 1
S  xS  (
xi )  (
i 0
n
 xi )
i 1
n 1
n
i 0
i 1
2S  (n  1)1
S  xS  1  ( xi )  ( xi  x n )
2S  (n  1)n
S (1  x)  1  x
i 1
(n  1)n
2
Sum of an Arithmetic Sequence
S
n
1 xn
,x  0
1 x
Sum of a Geometric Sequence
S
*The first two Sigma notations are the same,
but the bottom one is the top one going backwards.
Also, in general note the idea of how we've added or subtracted equations in both
these examples to essentially "get things to cross" out, so that we could simplify our
work and solve for a value we wanted to find. In general, the techniques of adding
equations, subtracting equations and even multiplying equations through by a value
(like we multiplied the equation for the geometric sequence through by x) are very
standard techniques to attempt to simplify things. There's no "formula" for when to do
these things, but the more examples you see, and the more you do problems, the
better feel you will get for when to try these things. Typically, I will try ideas like
these out and sometimes they will work and sometimes they won't. The important
thing is to have a set of different techniques to try. If one doesn't work, then try
something else.