1. **Stating the problem:** We are given the number of squares formed in patterns 1 and 2 and need to complete the table for patterns 3 to 6. 2. **Given data:** Pattern number: 1, 2, 3, 4, 5, 6 Number of squares: 5, 10, ?, ?, ?, ? 3. **Step 1: Find the number of squares for patterns 3 to 6.** We observe the pattern for the number of squares formed. Let's try to find a formula or sequence. 4. **Step 2: Use the general term form:** The number of squares in pattern $n$ is $T_n = n^2 + bn + c$, where $b, c \in \mathbb{N}$. 5. **Step 3: Write $T_1$ and $T_2$ in terms of $b$ and $c$:** $$ T_1 = 1^2 + b(1) + c = 1 + b + c $$ $$ T_2 = 2^2 + b(2) + c = 4 + 2b + c $$ 6. **Step 4: Substitute the known values:** From the table, $T_1 = 5$ and $T_2 = 10$. So, $$ 1 + b + c = 5 $$ $$ 4 + 2b + c = 10 $$ 7. **Step 5: Solve the system of equations:** Subtract the first equation from the second: $$ (4 + 2b + c) - (1 + b + c) = 10 - 5 $$ $$ 4 + 2b + c - 1 - b - c = 5 $$ $$ 3 + b = 5 $$ $$ b = 5 - 3 = 2 $$ 8. **Step 6: Find $c$ using $b=2$ in the first equation:** $$ 1 + 2 + c = 5 $$ $$ 3 + c = 5 $$ $$ c = 5 - 3 = 2 $$ 9. **Step 7: Write the general term:** $$ T_n = n^2 + 2n + 2 $$ 10. **Step 8: Calculate $T_3$, $T_4$, $T_5$, and $T_6$:** $$ T_3 = 3^2 + 2(3) + 2 = 9 + 6 + 2 = 17 $$ $$ T_4 = 4^2 + 2(4) + 2 = 16 + 8 + 2 = 26 $$ $$ T_5 = 5^2 + 2(5) + 2 = 25 + 10 + 2 = 37 $$ $$ T_6 = 6^2 + 2(6) + 2 = 36 + 12 + 2 = 50 $$ 11. **Step 9: Complete the table:** Pattern number: 1, 2, 3, 4, 5, 6 Number of squares: 5, 10, 17, 26, 37, 50 **Final answers:** - $b = 2$ - $c = 2$ - Number of squares for patterns 3 to 6 are 17, 26, 37, and 50 respectively.