1. **State the problem:** Find the sum of the arithmetic series (A.S) 25 + 21 + 17 + ... + (-23). 2. **Identify the first term ($a_1$), common difference ($d$), and last term ($a_n$):** - First term $a_1 = 25$ - Common difference $d = 21 - 25 = -4$ - Last term $a_n = -23$ 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: $$-23 = 25 + (n-1)(-4)$$ Simplify: $$-23 - 25 = (n-1)(-4)$$ $$-48 = -4(n-1)$$ Divide both sides by -4: $$12 = n - 1$$ $$n = 13$$ 4. **Calculate the sum ($S_n$) of the arithmetic series:** Use the sum formula: $$S_n = \frac{n}{2} (a_1 + a_n)$$ Substitute values: $$S_{13} = \frac{13}{2} (25 + (-23)) = \frac{13}{2} (2) = 13$$ **Final answer:** The sum of the arithmetic series is $13$.