1. **Stating the problem:** We have a sequence of dot patterns where each pattern number $n$ corresponds to a certain number of dots. The number of dots increases by the same amount each time. We want to find an expression for the number of dots in the $n^{th}$ pattern and then find how many dots are in the 14th pattern. 2. **Observing the given data:** - Pattern 1 has 5 dots. - Pattern 2 has 9 dots. - Pattern 3 has 14 dots. 3. **Finding the pattern:** Calculate the differences between consecutive terms: $$9 - 5 = 4$$ $$14 - 9 = 5$$ The differences are not constant, so it is not an arithmetic sequence. 4. **Check second differences:** Second difference: $$5 - 4 = 1$$ Since the second difference is constant, the sequence is quadratic. 5. **General quadratic formula:** $$a_n = An^2 + Bn + C$$ We need to find $A$, $B$, and $C$. 6. **Use the known values:** For $n=1$, $a_1=5$: $$A(1)^2 + B(1) + C = 5 \Rightarrow A + B + C = 5$$ For $n=2$, $a_2=9$: $$4A + 2B + C = 9$$ For $n=3$, $a_3=14$: $$9A + 3B + C = 14$$ 7. **Solve the system:** Subtract first equation from second: $$(4A + 2B + C) - (A + B + C) = 9 - 5$$ $$3A + B = 4$$ Subtract first equation from third: $$(9A + 3B + C) - (A + B + C) = 14 - 5$$ $$8A + 2B = 9$$ 8. **Simplify and solve:** From $3A + B = 4$, express $B$: $$B = 4 - 3A$$ Substitute into $8A + 2B = 9$: $$8A + 2(4 - 3A) = 9$$ $$8A + 8 - 6A = 9$$ $$2A + 8 = 9$$ $$2A = 1$$ $$A = \frac{1}{2}$$ Then, $$B = 4 - 3 \times \frac{1}{2} = 4 - \frac{3}{2} = \frac{5}{2}$$ From $A + B + C = 5$: $$\frac{1}{2} + \frac{5}{2} + C = 5$$ $$3 + C = 5$$ $$C = 2$$ 9. **Final formula:** $$a_n = \frac{1}{2}n^2 + \frac{5}{2}n + 2$$ 10. **Simplify formula:** Multiply numerator and denominator to write as: $$a_n = \frac{n^2 + 5n + 4}{2}$$ 11. **Calculate the 14th pattern:** $$a_{14} = \frac{14^2 + 5 \times 14 + 4}{2} = \frac{196 + 70 + 4}{2} = \frac{270}{2} = 135$$ **Answer:** - Expression for the number of dots in the $n^{th}$ pattern is: $$a_n = \frac{n^2 + 5n + 4}{2}$$ - Number of dots in the 14th pattern is: $$135$$