Method

Direct Proof

A direct proof is a proof that uses the given information and other known facts until the statement is shown to be true. Consider the following statement.

If $n$ is an odd number, then $n^{2}$ is also an odd number.

A direct proof is dependent on what information is available, and what is the statement to be proven. The following steps summarize, in general, how to do a direct proof.

Understand the Given Statement

The aim is to prove that given any odd number $n,$ its square $n^{2}$ is also odd. To better understand the statement some example cases can be worked.

$n$	$n^{2}$	Is $n^{2}$ odd?
$1$	$1$	Yes
$3$	$9$	Yes
$5$	$25$	Yes
$7$	$49$	Yes

In the above table, the statement was proven to be true for just a few odd numbers, but the goal is to prove that it is true for any odd number.

Reach the Conclusion Using Given Information, Known Facts, and Logical Reasoning

The given information and other known facts should be intertwined using logical reasoning several times. These can be other theorems, definitions, axioms, and so on. This step is repeated several times until a conclusion is reached.

Statement	Reason
$n$ is odd	Given.
$n = 2 k + 1$	Every odd number is equal to twice an integer plus $1 .$
$n^{2} = (2 k + 1)^{2}$	Raise the equation to the power of $2 .$
$n^{2} = 4 k^{2} + 4 k + 1$	Expand the square.
$n^{2} = 2 (2 k^{2} + 2 k) + 1$	Factor out $2 .$
$n^{2}$ is odd	It is written as twice an integer plus $1 .$

It is important to keep in mind that whether other theorems, definitions, or axioms are to be used or not depends on which type of statement is to be proven.

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