SOLUTION: the number of seats in the rows of a stadium form an arthemetic sequence. two employees of the stadium determine that the 13th row has 189 seats and the 25th row has 225 seats. how

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Question 1199392: the number of seats in the rows of a stadium form an arthemetic sequence. two employees of the stadium determine that the 13th row has 189 seats and the 25th row has 225 seats. how many seats are there in 55th row
Found 5 solutions by ikleyn, mccravyedwin, Edwin McCravy, greenestamps, math_tutor2020:
Answer by ikleyn(52748)   (Show Source): You can put this solution on YOUR website!
.
the number of seats in the rows of a stadium form an arthemetic sequence.
two employees of the stadium determine that the 13th row has 189 seats
and the 25th row has 225 seats. how many seats are there in 55th row
~~~~~~~~~~~~~~~~~ 

Between the 13-th row and 25-th row, there are 25-13 = 12 gaps,
and the increment of the number of seats is  225-189 = 36.


Thus each gap brings   = 3 additional seats.


So d= 3 is the common difference of the arithmetic progression .


Between the 25-th row and 55-th row, there are 55-25 = 30 gaps,
and each gap adds 3 seats; so the number of seats in the 55-th row is

    225 + 3*30 = 225 + 90 = 315.    ANSWER

Solved.



Answer by mccravyedwin(405)   (Show Source): You can put this solution on YOUR website!

13 -----> 189
25 -----> 225
55 ----->  x




Solve for x.

Edwin
Answer by Edwin McCravy(20054)   (Show Source): You can put this solution on YOUR website!
Maybe you're supposed to do it this way:








Solve that by substitution or elimination, then substitute
the values you get for a1 and d and n=55 in



Edwin
Answer by greenestamps(13195)   (Show Source): You can put this solution on YOUR website!


Just comments about the other responses you have received on this....

Tutor @ikleyn first presented an informal solution which is very easy to understand, with the only requirement being that you know what an arithmetic sequence is.

The other tutor then provides responses showing two ways to set up the problem for solving using formal algebra. They are both valid methods; but in the formal algebraic solutions it is not as easy to see WHY and HOW they work.

My comment, then, is this.

You should understand how the formal algebraic methods work, so you can apply them in other similar problems.

However, you will enjoy mathematics more if you have a clear understanding of how to solve the problem.

So know the formal algebraic solution methods; but if formal algebra is not required, go with the informal, common sense, easy-to-understand solution.
Answer by math_tutor2020(3816)   (Show Source): You can put this solution on YOUR website!

This is perhaps a slightly unconventional method when it comes to arithmetic sequences.

An arithmetic sequence is effectively a linear function with the domain of natural numbers {1,2,3,4,...}
As such, this is a discrete function rather than a continuous function.

A linear equation is of the form
y = mx+b

In this case, the x takes on values 1,2,3,...
and y is the number of seats in row x.

The 13th row has 189 seats. It translates to the notation (x,y) = (13,189)
In other words: x = 13 and y = 189 pair up together.
Another point is (25,225) since the 25th row has 225 seats.

Let's compute the slope
m = slope
m = rise/run
m = (y2-y1)/(x2-x1)
m = (225-189)/(25-13)
m = 36/12
m = 3
The slope indicates that each time x goes up by 1, y goes up by 3.

In this context, it means each row gets 3 extra seats compared to the previous row.

Let's use (x1,y1) = (13,189) along with that slope to determine the equation like so
y - y1 = m(x - x1)
y - 189 = 3(x - 13)
y - 189 = 3x - 39
y = 3x - 39 + 189
y = 3x + 150

That equation tells us how many seats are in row x, such that x is selected from the set {1,2,3,4,...}

For example, plug x = 13 into the equation to find that:
y = 3x + 150
y = 3*13 + 150
y = 39 + 150
y = 189
There are 189 seats in the 13th row.
This matches the info in the instructions.

Let's now try x = 25
y = 3x + 150
y = 3*25 + 150
y = 75 + 150
y = 225
This matches as well.
We have confirmed the correct linear equation.

The last thing to do is plug in x = 55 so we can find the value of y.
y = 3x + 150
y = 3*55 + 150
y = 165 + 150
y = 315

Answer: 315 seats
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