Question

Use the method of direct proof to prove the following statements. Every odd integer is a difference of two squares. (Example \(7=4^{2}-3^{2}\), etc.)

Short answer

Every odd number, expressed as \(2k+1\), can be rewritten as the difference of two squares in the form \((k+1)^2 - k^2\), and hence the statement is proved.

Step-by-step solution

Understand the concept of odd numbers

A key point to understand is the definition of an odd number. An odd integer is one that can be written in the form \(2k + 1\) where \(k\) is any integer.

Write the odd number as a difference of two squares

The expression \(2k + 1\) should be expressed as the difference of two squares. To do this, rewrite \(2k + 1\) as \((k+1)^2 - k^2 \). This still represents any odd integer, but is now also a difference of two squares.

Confirm the difference of squares

To confirm that \((k+1)^2 - k^2\) indeed always equals \(2k + 1\), expand the terms. On simplification, this will result in \(2k + 1\) proving that the approach is correct and applies to any integer \(k\).