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A perfect square is an important concept in mathematics. It refers to any number that can be expressed as \( k^2 \), where \( k \) is an integer. Essentially, this means multiplying an integer by itself. For example:

• \( 1^2 = 1 \)
• \( 2^2 = 4 \)
• \( 3^2 = 9 \)
• \( 4^2 = 16 \)

These numbers form a set of perfect squares: 1, 4, 9, 16, and so on.
Perfect squares are found regularly in both theoretical and practical math applications, helping to classify and identify non-square numbers.

In mathematics, an integer sequence is a sequence of numbers where each number is an integer. These sequences can follow different patterns, depending on the rule that defines them.
A well-known example is the sequence of perfect squares (1, 4, 9, 16, 25, ...), generated by squaring natural numbers (1, 2, 3, 4, 5, ...).
Another sequence is non-squares, forming when we take all natural numbers and remove all perfect squares from it.

• 1, 2, 3, 4, 5 evolves to 1, 2, 3, 5, 6
• 6, 7, 8, 9 evolves to 6, 7, 8, 10

Recognizing patterns in these sequences is essential for predicting the next numbers in the sequence and developing a formula.

Mathematics often simplifies complex processes using formulas. A formula is a concise way of expressing information symbolically, as in a mathematical equation. In the original exercise, the challenge was to find a formula for the \( n \)-th non-square number.
Here’s how you can think about it:

• Identify the number of perfect squares up to \( n \) using \( \lfloor \sqrt{n} \rfloor \).
• Subtract these from \( n \) to get the \( n \)-th non-square by accounting for missed non-squares due to these perfect squares.

Thus, the formula to approximate the \( n \)-th non-square can resemble \( n + \lfloor \sqrt{n} \rfloor \). This method ensures that by considering the number of perfect squares you bypass, a more accurate position of the \( n \)-th non-square is achieved.

The square root is a fundamental concept needed to solve many mathematical problems. The square root of a number \( x \) is a value that, when multiplied by itself, gives \( x \). Often, square roots are denoted as \( \sqrt{x} \).
For instance, the square root of 9 is 3, because \( 3 \times 3 = 9 \).

• The square root of 16 is 4, since \( 4 \times 4 = 16 \).

In the context of the exercise, knowing how to calculate the square root of a number allows you to count how many perfect squares are present up to a certain number \( n \). By using expressions like \( \lfloor \sqrt{n} \rfloor \), you get the highest integer less than or equal to this square root, effectively counting all perfect squares below or equal to \( n \).
Recognizing and using square roots is key in transitioning from understanding perfect squares to calculating non-squares.