The right side of the orchestra section of the Nederlander Theatre in New York City has 19 rows, and the last row has 27 seats. The number of seats in each row increases by 1 as you move toward the back of the section. How many seats are in this section of the theatre?
u are wrong my friend
To find the total number of seats in this section of the theatre, we need to find the sum of the seats in each row.
Since the number of seats in each row increases by 1 as you move toward the back of the section, it means that the first row will have the fewest seats, and the last row will have the most seats.
We are given that the last row has 27 seats. Let's denote the number of seats in the first row as "x". Since each row after the first row has 1 more seat than the previous row, the number of seats in the second row will be "x + 1", the number of seats in the third row will be "x + 2", and so on.
Since there are 19 rows in total, the number of seats in the 19th row will be "x + (19-1) = x + 18". Thus, we have the following pattern:
1st row: x seats
2nd row: x + 1 seats
3rd row: x + 2 seats
...
19th row: x + 18 seats
To find the total number of seats, we need to find the sum of all these row seat counts. This can be done by adding up each term in the pattern. Since there are 19 rows, we will have 19 terms in the sum.
The formula to find the sum of an arithmetic series (a series with a common difference) is given by:
Sum = (n/2)(2a + (n-1)d)
Here, n represents the number of terms (which is 19), a represents the first term (which is x), and d represents the common difference (which is 1 seat).
Substituting these values into the formula, we get:
Sum = (19/2)(2x + (19-1)1)
= (19/2)(2x + 18)
= 19(x + 9)
Now, we know that the total sum of seats in all the rows is equal to the total number of seats in the section, which is given to be 27 in the last row.
Therefore, we have the equation:
19(x + 9) = 27
To find the value of x, we can solve the equation:
19x + 171 = 27
19x = 27 - 171
19x = -144
x = -144/19
However, since the number of seats cannot be negative, we can conclude that there was an error in the problem or the information provided.
In this case, we cannot determine the exact number of seats in this section of the theatre without knowing the correct value of x.
this is just a simple arithmetic progression, with
a = 19
d = 1
Since there are 27-9+1=9 rows,
S9 = 9/2 (19+27) = 207