1. **State the problem:** We are given the quadratic sequence 9, 20, 33, 48, 65, ... and need to find the formula for the $n^{th}$ term. 2. **Identify the type of sequence:** Since it is quadratic, the $n^{th}$ term has the form $$a n^2 + b n + c$$ where $a$, $b$, and $c$ are constants. 3. **Find the first differences:** $$20 - 9 = 11$$ $$33 - 20 = 13$$ $$48 - 33 = 15$$ $$65 - 48 = 17$$ 4. **Find the second differences:** $$13 - 11 = 2$$ $$15 - 13 = 2$$ $$17 - 15 = 2$$ Since the second difference is constant and equals 2, this confirms the sequence is quadratic. 5. **Use the formula for the second difference:** The second difference equals $2a$, so $$2a = 2 \implies a = 1$$ 6. **Write the general term with $a=1$:** $$T_n = n^2 + b n + c$$ 7. **Use the first term to find $b$ and $c$:** For $n=1$, $$T_1 = 1^2 + b(1) + c = 1 + b + c = 9$$ So, $$b + c = 8$$ 8. **Use the second term:** For $n=2$, $$T_2 = 2^2 + 2b + c = 4 + 2b + c = 20$$ So, $$2b + c = 16$$ 9. **Solve the system:** Subtract the first equation from the second: $$ (2b + c) - (b + c) = 16 - 8$$ $$ \cancel{2b} + c - \cancel{b} - c = 8$$ $$b = 8$$ 10. **Find $c$:** From $b + c = 8$, $$8 + c = 8 \implies c = 0$$ 11. **Final formula:** $$T_n = n^2 + 8 n$$ **Answer:** The $n^{th}$ term rule for the sequence is $$\boxed{n^2 + 8 n}$$