Question: Fill in the missing terms for this arithmetic sequence.
□, -33, -19, □, □, ...
1. **State the problem:** We are given an arithmetic sequence with some missing terms: $$\square, -33, -19, \square, \square, ...$$
2. **Recall the formula for an arithmetic sequence:** The $n$-th term is given by $$a_n = a_1 + (n-1)d$$ where $a_1$ is the first term and $d$ is the common difference.
3. **Find the common difference $d$:** Using the 2nd and 3rd terms,
$$a_2 = -33, \quad a_3 = -19$$
Since $$a_3 = a_2 + d,$$
we have
$$-19 = -33 + d$$
$$d = -19 + 33 = 14$$
4. **Find the first term $a_1$:** Using $$a_2 = a_1 + d,$$
$$-33 = a_1 + 14$$
$$a_1 = -33 - 14 = -47$$
5. **Find the 4th and 5th terms:**
$$a_4 = a_1 + 3d = -47 + 3 \times 14 = -47 + 42 = -5$$
$$a_5 = a_1 + 4d = -47 + 4 \times 14 = -47 + 56 = 9$$
6. **Final sequence:**
$$-47, -33, -19, -5, 9, ...$$