Subjects algebra

Arithmetic Series Sum E36222

Step-by-step solutions with LaTeX - clean, fast, and student-friendly.

Use the AI math solver

1. **State the problem:** Find the sum of the arithmetic series $$\sum_{i=1}^{18} (14 - 6i)$$. 2. **Formula used:** The sum of an arithmetic series with $n$ terms, first term $a_1$, and last term $a_n$ is given by: $$S_n = \frac{n}{2}(a_1 + a_n)$$ 3. **Find the first term $a_1$:** Substitute $i=1$ into the expression: $$a_1 = 14 - 6(1) = 14 - 6 = 8$$ 4. **Find the last term $a_{18}$:** Substitute $i=18$: $$a_{18} = 14 - 6(18) = 14 - 108 = -94$$ 5. **Number of terms:** $n = 18$ 6. **Calculate the sum:** $$S_{18} = \frac{18}{2}(8 + (-94)) = 9 \times (8 - 94) = 9 \times (-86) = -774$$ **Final answer:** $$\boxed{-774}$$