There are 30 rows of seats in the North Wing of a stadium. Each row has 2 seats more than the row in front. The last row has 132 seats. How many seats does the first row have? How many seats are there altogether in the North Wing of the stadium?
the seats form an AP with
a, a+2, a+4, .... , 132
here d=2, n=30
term(30) = a + 29d = 132
a + 29(2) = 132
a = 74
sum of 30 rows = (n/2)( first + last)
= 15(74 + 132) = .....
or
use: Sum(n) = (n/2)(2a + (n-1)d)
To find the number of seats in the first row, we need to work backwards from the last row.
Let's assume the number of seats in the first row is 'x'.
The number of seats in the second row will be 'x + 2' because each row has 2 seats more than the row in front.
Similarly, the number of seats in the third row will be 'x + 4', in the fourth row will be 'x + 6', and so on.
Since the last row has 132 seats, we can set up the equation:
x + (30 - 1) * 2 = 132
Let's solve this equation:
x + 58 = 132
Subtracting 58 from both sides:
x = 132 - 58
x = 74
Therefore, the first row has 74 seats.
To find the total number of seats in the North Wing, we need to sum up the number of seats in each row.
The sum of an arithmetic series can be calculated using the formula:
sum = (first term + last term) * number of terms / 2
In this case, the first term is x = 74 (the number of seats in the first row), the last term is 132 (the number of seats in the last row), and the number of terms is 30 (the total number of rows).
Plugging these values into the formula, we get:
sum = (74 + 132) * 30 / 2
sum = 206 * 15
sum = 3090
Therefore, there are 3090 seats altogether in the North Wing of the stadium.
To find the number of seats in the first row, we need to determine the number of seats that are added to each row as we move towards the back.
Let's assume the number of seats in the first row is represented by "x". Since each row has 2 seats more than the row in front, the second row will have "x + 2" seats. The third row will have "x + 2 + 2" seats, which simplifies to "x + 4".
We can observe a pattern: each row has an additional 2 seats compared to the previous row. So, we can create an equation using this pattern:
x + (x + 2) + (x + 4) + ... + (x + 2(n-1))
In this equation, "n" represents the number of rows. In this case, n = 30.
Substituting 30 for n, we get:
x + (x + 2) + (x + 4) + ... + (x + 2(30-1))
Now, we need to sum the terms:
30x + 2(1 + 2 + 3 + ... + 29)
The sum of the series 1 + 2 + 3 + ... + n can be found using the formula n(n + 1) / 2. In our case, n = 29:
30x + 2(29(29 + 1) / 2)
Simplifying further:
30x + 2(29 * 30 / 2)
30x + 2(15 * 29)
30x + 2(435)
30x + 870
We also know that the last row has 132 seats, so we can set up another equation:
x + (30 - 1) * 2 = 132
Simplifying:
x + 58 = 132
x = 132 - 58
x = 74
Therefore, the first row has 74 seats.
To find the total number of seats in the North Wing of the stadium, we can plug the value of x back into the first equation:
30x + 870
30(74) + 870
2220 + 870
Therefore, there are 3090 seats altogether in the North Wing of the stadium.