Quadratic Sequence 9E2552

1. **State the problem:** We are given the first four terms of a quadratic sequence: $3$, $10$, $19$, $30$. We need to find a formula for the $n$th term, $T_n$, in terms of $n$. 2. **Recall the formula for a quadratic sequence:** The $n$th term of a quadratic sequence can be written as: $$T_n = an^2 + bn + c$$ where $a$, $b$, and $c$ are constants to be determined. 3. **Find the first differences:** $$10 - 3 = 7$$ $$19 - 10 = 9$$ $$30 - 19 = 11$$ So the first differences are $7$, $9$, and $11$. 4. **Find the second differences:** $$9 - 7 = 2$$ $$11 - 9 = 2$$ The second differences are constant and equal to $2$, confirming the sequence is quadratic. 5. **Use the second difference to find $a$:** The second difference equals $2a$, so: $$2a = 2 \\ a = 1$$ 6. **Set up equations using the first three terms:** For $n=1$: $$T_1 = a(1)^2 + b(1) + c = 1 + b + c = 3$$ For $n=2$: $$T_2 = a(2)^2 + b(2) + c = 4 + 2b + c = 10$$ For $n=3$: $$T_3 = a(3)^2 + b(3) + c = 9 + 3b + c = 19$$ 7. **Write the system of equations:** $$1 + b + c = 3 \\ 4 + 2b + c = 10 \\ 9 + 3b + c = 19$$ 8. **Simplify each equation:** $$b + c = 2 \\ 2b + c = 6 \\ 3b + c = 10$$ 9. **Subtract the first equation from the second:** $$(2b + c) - (b + c) = 6 - 2 \\ b = 4$$ 10. **Substitute $b=4$ into $b + c = 2$:** $$4 + c = 2 \\ c = 2 - 4 = -2$$ 11. **Write the final formula:** $$T_n = n^2 + 4n - 2$$ **Answer:** The $n$th term of the sequence is $$\boxed{n^2 + 4n - 2}$$.