1. **Stating the problem:** We have a sequence of T-shaped patterns made of square tiles each with side length 1 cm. - Pattern 1 has 5 tiles: a horizontal row of 3 tiles and a vertical column of 3 tiles sharing the middle tile. - Pattern 2 has 6 tiles: a horizontal row of 4 tiles and a vertical column of 3 tiles sharing the middle tile. - Pattern 3 has 7 tiles: a horizontal row of 5 tiles and a vertical column of 3 tiles sharing the middle tile. We need to find the perimeter of the $n^{\text{th}}$ pattern. 2. **Understanding the pattern:** - The vertical column always has 3 tiles. - The horizontal row has $n+2$ tiles (since pattern 1 has 3 tiles, pattern 2 has 4, pattern 3 has 5, etc.). - The middle tile is shared between the vertical and horizontal parts. 3. **Finding the dimensions of the shape:** - Height: The vertical column has 3 tiles, so height is $3$ cm. - Width: The horizontal row has $n+2$ tiles, so width is $n+2$ cm. 4. **Calculating the perimeter:** The shape is a T formed by a vertical column of height 3 and a horizontal row of width $n+2$ crossing at the middle tile. - The vertical column extends 3 tiles down. - The horizontal row extends $n+2$ tiles across. The perimeter is the total length around the shape. 5. **Visualizing the perimeter:** - The top horizontal row is $n+2$ tiles wide. - The vertical column extends 3 tiles down from the middle tile of the horizontal row. 6. **Calculating perimeter step-by-step:** - The top edge length is $n+2$ cm. - The bottom edge length is also $n+2$ cm. - The vertical edges on the sides are each 3 cm tall. However, the vertical column is centered under the horizontal row, so the shape looks like a T. 7. **Perimeter formula:** The perimeter $P$ is the sum of all outer edges: $$ P = \text{top edge} + \text{bottom edge} + 2 \times \text{vertical edges} + 2 \times \text{horizontal edges of vertical column} $$ But since the vertical column is 3 tiles tall and 1 tile wide, and the horizontal row is $n+2$ tiles wide and 1 tile tall, the perimeter is: $$ P = 2(n+2) + 2 \times 3 + 2 \times 1 = 2n + 4 + 6 + 2 = 2n + 12 $$ 8. **Final answer:** $$ \boxed{P = 2n + 12 \text{ cm}} $$ This formula gives the perimeter of the $n^{\text{th}}$ T-shaped pattern made of tiles with side length 1 cm.