Answer: A. 2,604
Step-by-step explanation:
According to the given information the number of sets in rows following arithmetic progression (in short form AP).
Where , 23rd term is 65.
Common difference = 65-63=2
nth term in AP: [tex]a_n=a+(n-1)d[/tex]
For n= 23 , we have [tex]a_{23}=65[/tex] , d= 2
[tex]23=a+(23-1)2\\\\ 23= a+(22)2\\\\ 23=a+44\\\\ a=44-23=21[/tex]
Formula for the sum of first n terms in AP= [tex]S_n=\dfrac{n}{2}[2a+(n-1)d][/tex]
Since, There are 42 rows in the auditorium.
Then,
[tex]S_{42}=\dfrac{42}{2}[2(21)+(42-1)2]\\\\=21(42+82)\\\\=21(124)=2604[/tex]
Hence, the the auditorium contains 2,604 seats.
Thus , the correct answer is A. 2,604
We want to find the total number of seats in the auditorium, knowing that the number of seats is given by an arithmetic sequence.
The correct option is A: there are 2,604 seats in total.
We know that in the row before you (the number 22) there are 63 seats, in your row (number 23) there are 65 seats, and in the next row there are 67 seats, then we can write the 3 terms of a sequence:
[tex]a_{22}= 63\\a_{23} = 65\\a_{24} = 67[/tex]
Is simple to see that this is an arithmetic sequence, such that the difference between consecutive terms is 2, then the number of seats in the n-th row is given by:
[tex]a_n = a_{n-1} + 2[/tex]
or:
[tex]a_n = a_1 + (n - 1)*2[/tex]
Using this second equation, we can find the value of the first term of the sequence:
[tex]a_{23} = 65 = a_1 + 2*(23 - 1)\\\\65 - 44 = a_1 = 21\\[/tex]
Now, if we define d as the difference between consecutive terms in an arithmetic sequence, the sum of the first N terms is given by:
[tex]S(N) = N*(2*a_1 + (N - 1)*d)/2[/tex]
Here we have a total of 42 rows, so we use N = 42.
d =2
a₁ = 21
Then the total number of seats is:
[tex]S(N) = 42*(2*21 + (42 - 1)*2)/2 = 2,604[/tex]
This means that the correct option is A: there are 2,604 seats in total.
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