An auditorium has 30 rows with 10 seats in the first row, 12 in the second row, 14 in the third row, and so forth. How many seats are in the auditorium?
looks like an arithmetic series where
a = 10, d=2 and n=30
sum(30) = (30/2)[2(10) + (29)(2)]
= 15[20+58] = 1170
(using the standard formula for the sum of n terms)
To find the total number of seats in the auditorium, we can calculate the sum of the seats in each row. In this case, we know that the number of seats in each row forms an arithmetic sequence, where the difference between consecutive terms is 2.
To determine the number of seats in each row, we can use the formula for the nth term of an arithmetic sequence:
a_n = a_1 + (n - 1)d
where:
a_n is the nth term in the sequence,
a_1 is the first term in the sequence,
n is the position of the term, and
d is the common difference between terms.
In this case, a_1 = 10 (the number of seats in the first row) and d = 2 (since the difference between consecutive rows is 2).
We can calculate the number of seats in each row using this formula. For example, to find the number of seats in the third row:
a_3 = 10 + (3 - 1)2
= 10 + 4
= 14
Similarly, we can find the number of seats in the remaining rows:
a_2 = 10 + (2 - 1)2
= 10 + 2
= 12
a_4 = 10 + (4 - 1)2
= 10 + 6
= 16
a_5 = 10 + (5 - 1)2
= 10 + 8
= 18
Continuing this pattern, we can find the number of seats in each row up to the 30th row.
To calculate the total number of seats, we can sum the number of seats in each row:
Total seats = a_1 + a_2 + a_3 + ... + a_30
Total seats = 10 + 12 + 14 + ... + a_30
To simplify the calculation, we can use the formula for the sum of the first n terms of an arithmetic sequence:
S_n = (n/2)(a_1 + a_n)
where:
S_n is the sum of the first n terms,
n is the number of terms to be added, and
a_n is the last term in the sequence.
In this case, n = 30 (since there are 30 rows) and a_n = a_30 (the number of seats in the 30th row).
Using the formula, we can calculate the total number of seats in the auditorium:
Total seats = (30/2)(10 + a_30)
Therefore, to find the total number of seats in the auditorium, we need to calculate a_30, the number of seats in the 30th row.