1. **State the problem:** We are given the quadratic sequence: -5, -2, 3, 10, 19, ... and need to find the formula for the $n^{th}$ term. 2. **Identify the pattern:** Quadratic sequences have a second difference that is constant. 3. **Calculate first differences:** $-2 - (-5) = 3$ $3 - (-2) = 5$ $10 - 3 = 7$ $19 - 10 = 9$ 4. **Calculate second differences:** $5 - 3 = 2$ $7 - 5 = 2$ $9 - 7 = 2$ Since the second difference is constant and equals 2, the sequence is quadratic. 5. **General form of quadratic sequence:** $$a n^2 + b n + c$$ 6. **Use the second difference to find $a$:** Second difference $= 2a = 2$ so $a = 1$ 7. **Set up equations using terms:** For $n=1$, term is $-5$: $$1(1)^2 + b(1) + c = -5 \Rightarrow 1 + b + c = -5$$ For $n=2$, term is $-2$: $$1(2)^2 + b(2) + c = -2 \Rightarrow 4 + 2b + c = -2$$ 8. **Solve the system:** From first: $$b + c = -6$$ From second: $$2b + c = -6$$ Subtract first from second: $$(2b + c) - (b + c) = -6 - (-6) \Rightarrow b = 0$$ 9. **Find $c$:** From $b + c = -6$ and $b=0$, we get $c = -6$ 10. **Final formula:** $$n^2 - 6$$ This is the $n^{th}$ term rule for the sequence. **Answer:** The $n^{th}$ term is $$n^2 - 6$$.