A theater has 30 seats in the first row, 33 seats in the second row, 36 seats in the third row, and so on in the same increasing pattern. If the theater has 15 rows of seats, how many seats are in the theater?
If the initial term of an arithmetic progression is a1 and the common difference of successive members is d, then the nth term of the sequence (an is given by:
an = a1 + ( n - 1 ) * d
This sum :
S = n * [ 2 a1 + ( n - 1 ) * d ] / 2
In this case :
a1 = 30
d = 3
n = 15
S = n * [ 2 a1 + ( n - 1 ) * d ] / 2
S = 15 * [ 2 * 30 + ( 15 - 1 ) * 3 ] / 2 =
15 * ( 60 + 14 * 3 ) / 2 =
15 * ( 60 + 42 ) / 2 =
15 * 102 / 2 = 1530 / 2 = 765
765 seats
how do you round to the nearest gram
To solve this problem, we need to find the total number of seats in each row and sum them up.
First, let's calculate the number of seats in each row using the given pattern.
For the first row, there are 30 seats.
For the second row, there are 33 seats.
For the third row, there are 36 seats.
And so on...
We can observe that the number of seats in each row increases by 3 with each subsequent row.
So, the number of seats in the fourth row would be 36 + 3 = 39.
Similarly, the number of seats in the fifth row would be 39 + 3 = 42.
And so on...
Now, we can create a sequence for the number of seats in each row: 30, 33, 36, 39, 42, ...
To find the number of seats in the 15th row, we need to find the 15th term of this sequence.
We can use the formula for arithmetic sequence to find the nth term:
an = a1 + (n - 1)d
where:
an is the nth term
a1 is the first term
n is the position of the term in the sequence
d is the common difference
In this case, a1 = 30, n = 15, and d = 3.
Let's calculate the 15th term:
a15 = 30 + (15 - 1)3
= 30 + 14 * 3
= 30 + 42
= 72
Therefore, the 15th row has 72 seats.
To find the total number of seats in the theater, we need to sum up the number of seats in each row from the first row to the 15th row.
To find the sum of an arithmetic series, we can use the formula:
Sn = (n/2)(a1 + an)
where:
Sn is the sum of the series
a1 is the first term
an is the nth term
n is the number of terms
In this case, a1 = 30, an = 72, and n = 15.
Let's calculate the sum:
Sn = (15/2)(30 + 72)
= (15/2)(102)
= 7.5 * 102
= 765
Therefore, there are a total of 765 seats in the theater.
To find the total number of seats in the theater, we can use the formula for the sum of an arithmetic sequence.
The formula for the sum of the first "n" terms of an arithmetic sequence is given by:
Sn = (n/2) * (a + l)
Where:
- Sn is the sum of the terms
- n is the number of terms
- a is the first term
- l is the last term
In this case, we know that there are 15 rows of seats, and the first row has 30 seats.
Using the given pattern, we can find the number of seats in the last row by adding the number of seats in each row. We observe that the number of seats in each row is increasing by 3.
So, the number of seats in the last row would be:
30 + 33 + 36 + ... (repeating the pattern 15 times)
This can be simplified by using the formula for the sum of an arithmetic series. The first term "a" of the series is 30, the difference "d" is 3, and the number of terms "n" is 15.
Sn = (n/2) * (2a + (n-1)d)
Substituting the values, we get:
S15 = (15/2) * (2*30 + (15-1)*3)
= (15/2) * (60 + 14*3)
= 15 * (60 + 42)
= 15 * 102
= 1530
Therefore, the theater has a total of 1530 seats.