1. **Stating the problem:** We are given the sequence 2, 8, 18, 32, 50 and asked to find the pattern of the terms. 2. **Look for a pattern:** Let's denote the $n$th term as $a_n$. 3. **Check differences:** Calculate the first differences: $$8-2=6, \quad 18-8=10, \quad 32-18=14, \quad 50-32=18$$ 4. **Check second differences:** Calculate the differences of the first differences: $$10-6=4, \quad 14-10=4, \quad 18-14=4$$ Since the second differences are constant (equal to 4), the sequence follows a quadratic pattern. 5. **General quadratic form:** $$a_n = An^2 + Bn + C$$ 6. **Use the first three terms to find $A$, $B$, and $C$:** \begin{align*} a_1 &= A(1)^2 + B(1) + C = A + B + C = 2 \\ a_2 &= 4A + 2B + C = 8 \\ a_3 &= 9A + 3B + C = 18 \end{align*} 7. **Solve the system:** Subtract first from second: $$4A + 2B + C - (A + B + C) = 8 - 2 \Rightarrow 3A + B = 6$$ Subtract second from third: $$9A + 3B + C - (4A + 2B + C) = 18 - 8 \Rightarrow 5A + B = 10$$ Subtract the two equations: $$(5A + B) - (3A + B) = 10 - 6 \Rightarrow 2A = 4 \Rightarrow A = 2$$ Plug $A=2$ into $3A + B = 6$: $$3(2) + B = 6 \Rightarrow 6 + B = 6 \Rightarrow B = 0$$ Plug $A=2$, $B=0$ into $A + B + C = 2$: $$2 + 0 + C = 2 \Rightarrow C = 0$$ 8. **Final formula:** $$a_n = 2n^2$$ 9. **Verification:** $$a_1 = 2(1)^2 = 2$$ $$a_2 = 2(2)^2 = 8$$ $$a_3 = 2(3)^2 = 18$$ $$a_4 = 2(4)^2 = 32$$ $$a_5 = 2(5)^2 = 50$$ All terms match the given sequence. **Answer:** The pattern of the $n$th term is $$a_n = 2n^2$$.