determine the seating capacity of an auditorium with 46 rows if there are 18 seats in the first row, 21 in the second, and 24 in the third and so on
The first row has 18 seats and the last row hae 18 + 3*45 = 171 seats
The average is 85.5 seats. Multiply that by 46 for the total: 3933.
There is another way to do it:
46*18 + 3*(1 + 2 + 3 + ...45)
= 828 + (3/2)(45*46)
= 828 + 3105 = 3933
Here, I used the rule that
(1+2+3+...N) = (N)(N+1)/2
To determine the seating capacity of an auditorium with 46 rows, we need to calculate the number of seats in each row and then multiply it by the total number of rows.
In this case, the seating pattern is increasing by 3 seats per row. The first row has 18 seats, the second row has 21 seats, and so on.
We can observe that the number of seats in each row follows an arithmetic sequence with a common difference of 3.
To find the number of seats in each row, we can use the formula for the n-th term of an arithmetic sequence:
an = a1 + (n - 1)d
Where:
an represents the n-th term
a1 represents the first term
n represents the term number
d represents the common difference
Using this formula, we can calculate the number of seats in each row:
a1 = 18 (number of seats in the first row)
d = 3 (common difference)
For the 46th row, n = 46.
a46 = 18 + (46 - 1) * 3
a46 = 18 + 45 * 3
a46 = 18 + 135
a46 = 153
Therefore, the 46th row has 153 seats.
To find the seating capacity, we need to sum up the number of seats in each row. We can use the formula for the sum of an arithmetic series:
Sn = (n/2)(a1 + an)
Where:
Sn represents the sum of the series
n represents the number of terms
a1 represents the first term
an represents the last term
In this case, the number of rows is n = 46, and the first and last terms are a1 = 18 and an = 153, respectively.
Using this formula, we can calculate the seating capacity:
S46 = (46/2)(18 + 153)
S46 = (23)(171)
S46 = 3933
Therefore, the seating capacity of the auditorium with 46 rows is 3,933 seats.