1. **State the problem:** We have a sequence of patterns made from tiles. Each pattern adds the same number of tiles as the previous one. We want to find an expression for the number of tiles in the $n^{th}$ pattern. 2. **Analyze the given data:** - Pattern 1 has 4 tiles. - Pattern 2 has 8 tiles. - Pattern 3 has 14 tiles. 3. **Find the differences between consecutive terms:** - From 4 to 8: increase of $8 - 4 = 4$ tiles. - From 8 to 14: increase of $14 - 8 = 6$ tiles. 4. **Check if the increase is constant:** The increases are 4 and 6, which are not constant, so the sequence is not arithmetic. 5. **Check the second differences:** - Difference between 6 and 4 is $6 - 4 = 2$. Since the second difference is constant (2), the sequence follows a quadratic pattern. 6. **Assume the number of tiles in the $n^{th}$ pattern is:** $$T_n = an^2 + bn + c$$ 7. **Use the known values to form equations:** - For $n=1$: $a(1)^2 + b(1) + c = 4$ \Rightarrow $a + b + c = 4$ - For $n=2$: $4a + 2b + c = 8$ - For $n=3$: $9a + 3b + c = 14$ 8. **Solve the system:** - Subtract first from second: $(4a + 2b + c) - (a + b + c) = 8 - 4$ \Rightarrow $3a + b = 4$ - Subtract second from third: $(9a + 3b + c) - (4a + 2b + c) = 14 - 8$ \Rightarrow $5a + b = 6$ 9. **Subtract the two new equations:** $ (5a + b) - (3a + b) = 6 - 4$ \Rightarrow $2a = 2$ \Rightarrow $a = 1$ 10. **Find $b$:** From $3a + b = 4$, substitute $a=1$: $$3(1) + b = 4 \Rightarrow b = 1$$ 11. **Find $c$:** From $a + b + c = 4$, substitute $a=1$, $b=1$: $$1 + 1 + c = 4 \Rightarrow c = 2$$ 12. **Final expression:** $$T_n = n^2 + n + 2$$ **Answer:** The number of tiles in the $n^{th}$ pattern is given by $$\boxed{T_n = n^2 + n + 2}$$