1. **Problem statement:** We have a sequence representing the number of seats in each row of a hall. The first row has 9 seats, the second has 14 seats, the third has 19 seats, and so on. We want to find the total number of seats in 17 rows. 2. **Identify the sequence type:** The number of seats increases by 5 each row (14 - 9 = 5, 19 - 14 = 5), so this is an arithmetic sequence. 3. **Arithmetic sequence formula:** 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. 4. **Calculate the 17th term:** $$a_{17} = 9 + (17-1) \times 5 = 9 + 16 \times 5 = 9 + 80 = 89$$ 5. **Sum of the first $n$ terms:** The sum $S_n$ of an arithmetic sequence is $$S_n = \frac{n}{2} (a_1 + a_n)$$ 6. **Calculate the total seats:** $$S_{17} = \frac{17}{2} (9 + 89) = \frac{17}{2} \times 98 = 17 \times 49 = 833$$ 7. **Answer:** The total number of seats in the 17 rows is **833**. **Explanation of where the 2 comes from:** The 2 in the denominator of the sum formula comes from the formula for the sum of an arithmetic series, which averages the first and last terms and multiplies by the number of terms. It is a standard part of the formula: $$S_n = \frac{n}{2} (a_1 + a_n)$$ This divides by 2 to find the average of the first and last terms before multiplying by the number of terms.