This problem requires observing the pattern in the number of squares and then creating a sequence based on the term number.

Step 1: Analyze the Pattern Looking at the given shapes, we can observe how the pattern grows:

1. The first term has 5 squares.
2. The second term has 9 squares.
3. The third term has 13 squares.

The pattern adds 4 squares for each subsequent term. So the pattern is increasing by 4 squares per term.

Step 2: Create a Table of Values We can complete the table by observing that each term $T_n$ is defined by adding 4 to the previous term, starting with 5:

• Term 1: 5 squares
• Term 2: 9 squares
• Term 3: 13 squares
• Term 4: 17 squares
• Term 5: 21 squares
• Term 6: 25 squares

Step 3: Establish a Rule The pattern follows an arithmetic sequence where each term adds 4 to the previous one. The rule for the $n$-th term can be written as: $T_n = 4n + 1$ where $T_n$ represents the number of squares in the $n$-th term.

Would you like more details or have any questions?

Questions to Explore:

1. How can we determine the formula for any arithmetic sequence?
2. What if the pattern had different growth, like doubling each time? How would that affect the rule?
3. Can we use the formula $T_n = 4n + 1$ to find the 10th term?
4. What changes if the initial number of squares (starting term) is different?
5. How does this pattern relate to linear functions?

Tip: In arithmetic sequences, identifying the initial term and common difference allows you to form a general formula easily.