Answer:
There are 1645 seats in the theatre.
Step-by-step explanation:
The first row has 13 seats and each row has 2 seats more than the previous one which means
13,15,17...
As the terms are increasing with the same difference, d=2 this forms an arithmetic sequence.
where
[tex]d = 2\\a_1 = 13\\a_2 = 15\\a_3 = 17[/tex]
The rule for an arithmetic sequence is given by:
[tex]a_n = a_1 + (n-1)d[/tex]
Putting the value of a1 and d
[tex]a_n = 13+(n-1)(2)\\a_n = 13+2n-2\\a_n = 11+2n[/tex]
We have to find total seats in the theatre for that the 35th term of sequence is required as their are 35 rows.
So,
Putting n=35
[tex]a_{35} = 11+2(35)\\= 11+70\\=81[/tex]
Now we need to find the sum of the 35 terms
The formula for sum of arithmetic sequence is:
[tex]S_n = \frac{n}{2}(a_1+a_n)\\putting\ n=35\\S_{35} = \frac{35}{2}(a_1+a_{35})\\S_{35} = 17.5(13+81)\\S_{35} = 17.5(94)\\S_{35} = 1645[/tex]
Hence,
There are 1645 seats in the theatre.
