Let's consider the first diagram which is a \(3\times 3\) square, broken down into different sections.
Notice that this square is broken down into a \(2\times 2\) square, two \(2 \times 1\) rectangles, and a \(1 \times 1\) square in the top right corner. This shows one way to get a \(3 \times 3\) square from a \(2 \times 2\) square by adding \(5\) extra unit squares. Using areas, this diagram shows that
\[\begin{align*} 3 \times 3 &= 2 \times 2 + 1 \times 2 + 1 \times 2 + 1\\ & = 2 \times 2 + (2 \times 2 + 1) \end{align*}\]Rearranging this equation, and introducing exponents, we have that \(3^{2} - 2^{2} = 2 \times 2 + 1 = 5\).
Looking at the second diagram, we see a similar decomposition of a \(4 \times 4\) square into a \(3 \times 3\) square, two \(3 \times 1\) rectangles, and a \(1\times 1\) square.
We need \(7\) extra unit squares in this case. Using areas, this diagram shows that\[\begin{align*} 4 \times 4 &= 3 \times 3 + 1 \times 3 + 1 \times 3 + 1\\ & = 3 \times 3 + (2 \times 3 + 1) \end{align*}\]
It follows that \(4^{2} - 3^{2} = 2 \times 3 + 1 = 7\).
Can you see how the third diagram, and an area argument, gives us the following equality?
\[5^{2} - 4^{2} = 2 \times 4 + 1 = 9\]
In general, to get from a square of side length \(n\) to a square of side length \((n+1)\), we need to add \(2 \times n + 1\) unit squares. Can you explain why? This tells us something about the difference between consecutive square numbers. Notice that the expression \(2n+1\) generates the sequence of odd numbers \(3,~5,~7,~9,\ldots\) when we substitute the values \(n =1, ~2,~ 3,~ 4, \dots\).