The question asks for a formula to determine the number of square tiles in the $n$-th term of a sequence based on a series of figures composed of square tiles.

Looking at the provided image, the first four terms $a_1$, $a_2$, $a_3$, and $a_4$ are given:

• $a_1$ consists of 1 square.
• $a_2$ consists of 4 squares.
• $a_3$ consists of 7 squares.
• $a_4$ consists of 10 squares.

This sequence increases by 3 squares for each successive figure: $a_2 - a_1 = 3, \quad a_3 - a_2 = 3, \quad a_4 - a_3 = 3$ Thus, the sequence appears to follow an arithmetic pattern, where each term increases by 3.

To find a formula for the number of square tiles in the $n$-th term, we can note that this is an arithmetic sequence with a common difference $d = 3$ and the first term $a_1 = 1$.

The general formula for the $n$-th term of an arithmetic sequence is given by: $a_n = a_1 + (n-1) \cdot d$ Substituting $a_1 = 1$ and $d = 3$, we get: $a_n = 1 + (n-1) \cdot 3 = 1 + 3n - 3 = 3n - 2$ Thus, the correct formula to determine the number of square tiles in the $n$-th term is: $a_n = 3n - 2$

Let me know if you'd like further details or have any questions!

Here are five related questions:

1. How can we derive the general formula for an arithmetic sequence?
2. What is the importance of the common difference in an arithmetic sequence?
3. How can we verify that the formula $a_n = 3n - 2$ works for any term $n$?
4. How would the formula change if the sequence had a different common difference?
5. Can we visualize this sequence in a different way, such as plotting it on a graph?

Tip: Always verify your formula by plugging in known values (like $n = 1$, $n = 2$) to ensure it matches the given terms of the sequence.