1. **Stating the problem:** We have a sequence of patterns made of unit squares forming rectangles. The first four patterns have dimensions and perimeters as follows: - Pattern 1: 1 column × 2 rows, perimeter = 6 units - Pattern 2: 2 columns × 3 rows, perimeter = 10 units - Pattern 3: 3 columns × 4 rows, perimeter = ? units - Pattern 4: 4 columns × 5 rows, perimeter = ? units We need to: (a)(i) Draw the next pattern (Pattern 4) which is a 4×5 rectangle. (a)(ii) Complete the perimeter table for patterns 3, 4, and 5. (b)(i) Find a formula for the perimeter of pattern $n$. 2. **Understanding the pattern:** Each pattern $n$ is a rectangle with dimensions: $$\text{columns} = n$$ $$\text{rows} = n + 1$$ 3. **Formula for perimeter of a rectangle:** $$P = 2(\text{length} + \text{width})$$ Here, length = number of columns = $n$, width = number of rows = $n+1$. 4. **Calculate perimeter for pattern 3:** $$P_3 = 2(n + (n+1)) = 2(3 + 4) = 2 \times 7 = 14$$ 5. **Calculate perimeter for pattern 4:** $$P_4 = 2(4 + 5) = 2 \times 9 = 18$$ 6. **Calculate perimeter for pattern 5:** $$P_5 = 2(5 + 6) = 2 \times 11 = 22$$ 7. **Complete the table:** | Pattern number (n) | Perimeter (units) | |--------------------|-------------------| | 1 | 6 | | 2 | 10 | | 3 | 14 | | 4 | 18 | | 5 | 22 | 8. **Check if the perimeters form an arithmetic sequence:** Differences between consecutive perimeters: $$10 - 6 = 4$$ $$14 - 10 = 4$$ $$18 - 14 = 4$$ $$22 - 18 = 4$$ The common difference $d = 4$. 9. **Find the formula for the perimeter $P_n$ of pattern $n$:** The first term $a_1 = 6$, common difference $d = 4$. Arithmetic sequence formula: $$P_n = a_1 + (n-1)d = 6 + (n-1)4 = 6 + 4n - 4 = 4n + 2$$ 10. **Final formula:** $$\boxed{P_n = 4n + 2}$$ This formula gives the perimeter of pattern $n$ in units.