The seats are arranged in an arithmetic sequence, and the problem is asking you to find the sum of the arithmetic sequence.
The basic equation for an arithmetic sequence is:
[tex]a_{n} =a_{1}+(n-1)d \\ where\:a_{n}=n^{th}\:term\:of\:the\:sequence,\:a_{1}=first\:term\:of\:the\:sequence (11)[/tex]
n=# term, d=difference (8)
Plug in your values to get the equation:
[tex]a_{n} =11+(n-1)8 [/tex]
This equation satisfied the problem.
Now find the sum of the arithmetic sequence using the equation:
[tex] S_{n} = \frac{n}{2} (a_{1} + a_{n})[/tex]
[tex]where S_{n} = sum, \: n=number of terms.[/tex]
To substitute in values of [tex]a_{n}[/tex] and [tex]a_{1}[/tex] into the sum equation, they must be solved for first. n=29 because there are 29 rows.:
[tex]a_{1} =11+(1-1)8 = 11+0 = 11[/tex]
[tex]a_{29} =11+(29-1)8 = 11+ 224= 235[/tex]
Now substitute number into the sum equation and solve for S:
[tex] S_{n} = \frac{n}{2} (a_{1} + a_{n})[/tex]
[tex]S_{29} = \frac{29}{2} (11 + 235)[/tex]
[tex]S_{29} = \frac{29}{2} (11 + 235) = 3,567 [/tex]
The answer is 3,567 total seats in the auditorium.
