1. **State the problem:** We are given the first five terms of a quadratic sequence: 4, -3, -16, -35, -60. We need to find an expression for the nth term of the sequence in terms of $n$. 2. **Recall the formula for the nth term of a quadratic sequence:** The nth term of a quadratic sequence can be written as $$a_n = An^2 + Bn + C$$ where $A$, $B$, and $C$ are constants to be determined. 3. **Find the first differences:** Calculate the differences between consecutive terms: $$-3 - 4 = -7$$ $$-16 - (-3) = -13$$ $$-35 - (-16) = -19$$ $$-60 - (-35) = -25$$ So the first differences are: $-7, -13, -19, -25$. 4. **Find the second differences:** Calculate the differences between the first differences: $$-13 - (-7) = -6$$ $$-19 - (-13) = -6$$ $$-25 - (-19) = -6$$ The second differences are constant and equal to $-6$. 5. **Use the second difference to find $A$:** For a quadratic sequence, the second difference equals $2A$. So, $$2A = -6 \implies A = -3$$ 6. **Set up equations to find $B$ and $C$:** Using the general term $a_n = -3n^2 + Bn + C$, plug in the first three terms: - For $n=1$, $a_1 = 4$: $$-3(1)^2 + B(1) + C = 4 \implies -3 + B + C = 4 \implies B + C = 7$$ - For $n=2$, $a_2 = -3$: $$-3(2)^2 + B(2) + C = -3 \implies -12 + 2B + C = -3 \implies 2B + C = 9$$ 7. **Solve the system of equations:** From $B + C = 7$ and $2B + C = 9$: Subtract the first from the second: $$(2B + C) - (B + C) = 9 - 7 \implies B = 2$$ Substitute $B=2$ into $B + C = 7$: $$2 + C = 7 \implies C = 5$$ 8. **Write the nth term formula:** $$a_n = -3n^2 + 2n + 5$$ **Final answer:** The nth term of the sequence is $$\boxed{a_n = -3n^2 + 2n + 5}$$