This problem involves finding the general $n$-th term of a quadratic sequence. Let’s solve it step by step.

Sequence: $-6, -3, 2, 9, 18, \dots$

Step 1: Find the first differences.

$\text{First differences: } (-3 - (-6)), (2 - (-3)), (9 - 2), (18 - 9)$ $= 3, 5, 7, 9$

Step 2: Find the second differences.

$\text{Second differences: } (5 - 3), (7 - 5), (9 - 7)$ $= 2, 2, 2$

Since the second differences are constant, this confirms the sequence is quadratic.

Step 3: General form of a quadratic sequence.

The $n$-th term of a quadratic sequence is given by: $T_n = an^2 + bn + c$ We need to determine the coefficients $a$, $b$, and $c$ by substituting values for $n$ and $T_n$.

Step 4: Solve for $a$, $b$, and $c$.

Using the terms:

• For $n = 1, T_1 = -6$: $a(1)^2 + b(1) + c = -6 \quad \Rightarrow \quad a + b + c = -6$

• For $n = 2, T_2 = -3$: $a(2)^2 + b(2) + c = -3 \quad \Rightarrow \quad 4a + 2b + c = -3$

• For $n = 3, T_3 = 2$: $a(3)^2 + b(3) + c = 2 \quad \Rightarrow \quad 9a + 3b + c = 2$

This gives us the system of equations:

1. $a + b + c = -6$
2. $4a + 2b + c = -3$
3. $9a + 3b + c = 2$

Step 5: Solve the system of equations.

• Subtract (1) from (2): $(4a + 2b + c) - (a + b + c) = -3 - (-6)$ $3a + b = 3 \quad \text{(Equation 4)}$

• Subtract (2) from (3): $(9a + 3b + c) - (4a + 2b + c) = 2 - (-3)$ $5a + b = 5 \quad \text{(Equation 5)}$

• Subtract (4) from (5): $(5a + b) - (3a + b) = 5 - 3$ $2a = 2 \quad \Rightarrow \quad a = 1$

• Substitute $a = 1$ into (4): $3(1) + b = 3 \quad \Rightarrow \quad b = 0$

• Substitute $a = 1$ and $b = 0$ into (1): $1 + 0 + c = -6 \quad \Rightarrow \quad c = -7$

Step 6: Write the $n$-th term formula.

$T_n = an^2 + bn + c \quad \Rightarrow \quad T_n = n^2 - 7$

Final Answer:

The $n$-th term rule for the sequence is: $T_n = n^2 - 7$

Let me know if you'd like further clarifications! Here are some related questions for practice:

1. How would you check if a given sequence is quadratic?
2. Can you determine the $n$-th term formula for the sequence $2, 6, 12, 20, 30, \dots$?
3. What happens to the $n$-th term formula if the second differences are not constant?
4. How can you verify your answer for the $n$-th term using substitution?
5. What is the difference between an arithmetic sequence and a quadratic sequence?

Tip: For quadratic sequences, always confirm the second differences are constant before proceeding.