1. **State the problem:** Manny's Pizza has monthly net profits following an arithmetic sequence: 4, 1, -2, -5, ..., -56 (in hundreds of dollars). We need to find the total profit over this period. 2. **Identify the arithmetic sequence parameters:** - First term $a_1 = 4$ - Common difference $d = 1 - 4 = -3$ - Last term $a_n = -56$ 3. **Find the number of terms $n$:** Use the formula for the $n$th term of an arithmetic sequence: $$a_n = a_1 + (n-1)d$$ Substitute known values: $$-56 = 4 + (n-1)(-3)$$ Simplify: $$-56 = 4 - 3(n-1)$$ $$-56 - 4 = -3(n-1)$$ $$-60 = -3(n-1)$$ Divide both sides by $-3$: $$\cancel{-60} \div \cancel{-3} = \cancel{-3}(n-1) \div \cancel{-3}$$ $$20 = n - 1$$ $$n = 21$$ 4. **Calculate the total profit (sum of the arithmetic sequence):** The sum of $n$ terms is: $$S_n = \frac{n}{2}(a_1 + a_n)$$ Substitute values: $$S_{21} = \frac{21}{2}(4 + (-56)) = \frac{21}{2}(-52)$$ Calculate: $$S_{21} = 21 \times (-26) = -546$$ 5. **Interpret the result:** The sum is in hundreds of dollars, so total profit is: $$-546 \times 100 = -54600$$ **Final answer:** The total profit over this period is -54600.