A theater has 26 rows with 29 seats in the first row, 32 in the second row, 35 in the third row and so forth. How many seats are in the theater?

To find the total number of seats in the theater, we need to add up the number of seats in each row. We can use the formula for an arithmetic sequence to find the number of seats in each row: {eq}a_n = a_1 + (n-1)d {/eq} where {eq}a_n {/eq} is the nth term in the sequence, {eq}a_1 {/eq} is the first term, and {eq}d {/eq} is the common difference. In this case, the first term is 29 and the common difference is 3 (since each row has 3 more seats than the previous row). So, the formula for the number of seats in the nth row is: {eq}a_n = 29 + (n-1)3 {/eq} To find the total number of seats, we need to sum up the number of seats in each row from the first row to the 26th row: {eq}begin{aligned} Total number of seats &= sum_{n=1}^{26} a_n \ &= 29 + (29+3) + (29+2times3) +\cdots + (29+25times3) \ &= sum_{n=1}^{26} (29+3(n-1)) \ &= sum_{n=1}^{26} (3n+26) \ &= 3sum_{n=1}^{26} n + 26sum_{n=1}^{26} 1 \ &= 3cdotfrac{26cdot27}{2} + 26cdot26 \ &= 2,!296 end{aligned} {/eq} Therefore, there are 2,296 seats in the theater.