1. **State the problem:** We are given the first 5 terms of a quadratic sequence: 4, 11, 20, 31, 44. We need to find an expression for the nth term of this quadratic sequence. 2. **Identify the type of sequence:** A quadratic sequence has a general nth term of the form: $$a_n = An^2 + Bn + C$$ where $A$, $B$, and $C$ are constants to be determined. 3. **Find the first differences:** $$11 - 4 = 7$$ $$20 - 11 = 9$$ $$31 - 20 = 11$$ $$44 - 31 = 13$$ 4. **Find the second differences:** $$9 - 7 = 2$$ $$11 - 9 = 2$$ $$13 - 11 = 2$$ Since the second difference is constant and equals 2, this confirms the sequence is quadratic and: $$2A = 2 \implies A = 1$$ 5. **Use the formula for the nth term:** $$a_n = n^2 + Bn + C$$ 6. **Use known terms to find B and C:** For $n=1$, $a_1 = 4$: $$1^2 + B(1) + C = 4 \implies 1 + B + C = 4 \implies B + C = 3$$ For $n=2$, $a_2 = 11$: $$2^2 + B(2) + C = 11 \implies 4 + 2B + C = 11 \implies 2B + C = 7$$ 7. **Solve the system:** Subtract the first equation from the second: $$ (2B + C) - (B + C) = 7 - 3 \implies B = 4$$ Substitute $B=4$ into $B + C = 3$: $$4 + C = 3 \implies C = -1$$ 8. **Final expression:** $$a_n = n^2 + 4n - 1$$ **Answer:** The nth term of the quadratic sequence is: $$\boxed{a_n = n^2 + 4n - 1}$$