Question

An outdoor amphitheater has 100 seats in the front row and each subsequent row has 6 more seats than the previous row. how many seats are in in the amphitheater if there are 23 rows?

Answer 1

If each row have 6 more seats than the previous, then there is a common difference of 6 seats. This, in turn, becomes an arithmetic progression, with a₁ being 100 and d being 6. Now, we want to know the total number of seats when n = 23, so we need to use the sum of an arithmetic progression form:

$S_n = \frac{n}{2}(2a_1 + (n - 1)d)$
$S_{23} = \frac{23}{2}(2(100) + 22 \cdot 6)$
$S_{23} = \frac{23}{2}(200 + 132)$
$S_{23} = 3818$

Hence, there are 3818 seats in the first 23 rows.