a hockey arena has a total seating capacity of 15690.the first row of seats around the rink has 262 seats. the number of seats in each subsequent row increases by 18. how many rows of seats are in the arena?

Solve using aritmetic progression and quadratic equation, how?

Sum of AP = n/2(2a+(n-1)d)
Where n is the number of terms, ie in your case the number of rows of seats, 'a' is the number in the first row (a=262) and 'd' is the difference in the next row (d=18). Sum of AP is the number of seats in the arena (Sum=15690). Put this together and you get:
15690=n/2[2*262+(n-1)18]
18n^2+506n-31380=0
equation to: ax^2+bx+c,; a=18, b=506, c=-31380
Now comes the quadratic, solution to a quadratic =
[-b­(+/-)Sqrt(b^2-4ac)]/2a
substitute the values of a,b anc from above into the quadratic and solve.
n=30.
Please always check your answer, in this case use sum of AP.

To find the number of rows of seats in the arena, we need to divide the total number of seats by the number of seats in each subsequent row.

The first row has 262 seats, and each subsequent row has 18 more seats. This means the second row has 262 + 18 = 280 seats, the third row has 280 + 18 = 298 seats, and so on.

Let's create an equation to represent this pattern:

Total number of seats = Number of seats in the first row + Number of seats in the second row + ... + Number of seats in the last row

Total number of seats = 262 + (262 + 18) + (262 + 2 * 18) + ...

We can see that the difference between each term in the sequence is 18, and the first term is 262. To find the number of rows, we need to find the last term in the sequence.

The last term can be found using the formula for the nth term of an arithmetic sequence:

Last term = First term + (n - 1) * Common difference

15690 = 262 + (n - 1) * 18

Now, let's solve for n:

15690 - 262 = (n - 1) * 18
15428 = 18n - 18
15428 + 18 = 18n
15446 = 18n
n = 15446 / 18
n ≈ 857.44

Since the number of rows must be a whole number, we can round down to the nearest whole number.

Therefore, there are 857 rows of seats in the arena.

To find the number of rows of seats in the arena, we need to determine how many times the seating capacity increases by 18 until we reach or exceed the total seating capacity of 15,690.

Let's start by finding the total number of seats in the first row.
Given that the first row has 262 seats, we can denote this as S1 = 262.

Now, let's calculate the total number of seats in the second row.
Since each subsequent row increases by 18 seats, we can denote the number of seats in the second row as S2 = S1 + 18.

Continuing this pattern, we can find the number of seats in the third row as S3 = S2 + 18, and so on.

To determine the number of rows, we need to keep adding 18 seats until we reach or exceed the total seating capacity of 15,690.

Let's perform the calculations:

S1 = 262
S2 = S1 + 18 = 262 + 18 = 280
S3 = S2 + 18 = 280 + 18 = 298
...
Sn = S(n-1) + 18

We need to find the value of n when Sn reaches or exceeds 15,690.

To simplify the calculation, we can use a loop or an equation.

Let's use an equation to find the number of rows:

Sn = S1 + (n-1) * 18

We know that the total number of seats in the arena is 15,690.

So, 15690 = 262 + (n-1) * 18

Rearranging the equation, we get:
(n-1) * 18 = 15690 - 262
(n-1) * 18 = 15428

Now, let's solve for n by dividing both sides of the equation by 18:
(n-1) = 15428 / 18
(n-1) = 857.11

We want to find a whole number of rows, so let's round up the decimal value to the nearest whole number.

(n-1) ≈ 857.11 (rounding up)
n - 1 ≈ 858
n ≈ 859

Therefore, there are approximately 859 rows of seats in the arena.