An auditorium has 49 rows of seats. The first row contains 40 seats. As you move to the rear of the auditorium, each row has 3 more seats than the previous row.
* I am pretty sure it involves both arithmetic series and explicit formulas but im not sure exactly how to go about it
OK. First, what exactly is the question? How many seats in the last row? How many seats in the auditorium?
Let's figure a model to start.
Row 1: 40
Row 2: 43
Row 3: 46
Row 4: 49
and so on.
We want a formula that, given the row, will tell us how many seats.
Multiplying the row by 3 has to be part of the answer, since there are 3 extra in each row, but that doesn't account for the initial row.
We _could_ subtract 3 from the 40, imagining a row 0 with 37 seats that doesn't exist. Then our formula would be nice and neat:
Seats in row = 37 + row * 3
Or we could equally well start at 40 but subtract 1 from the number of rows, to get:
Seats in row = 40 + (row - 1) * 3
I'll use the first.
s = 3r + 37
So in row 49 we have 3*49 + 37 seats.
you have 49 rows. row 1 has 40 seats. all u do is 40(n). 49*3=147 + 40= 187
To find the number of seats in each row as you move towards the rear, you can use arithmetic series and explicit formulas. Here's how you can proceed:
1. Determine the common difference (d) between the number of seats in each row. Since each row has 3 more seats than the previous row, the common difference is 3.
2. Identify the first term (a_1) - this is the number of seats in the first row, which is given as 40.
3. Calculate the explicit formula for the number of seats in the nth row (a_n) in terms of n using the formula:
a_n = a_1 + (n - 1)d,
where a_n represents the number of seats in the nth row, a_1 is the first term, n is the row number, and d is the common difference.
In this case, the explicit formula is:
a_n = 40 + (n - 1)3.
4. To find the total number of seats in the auditorium, you'll need to find the sum of the number of seats in all the rows. This can be done using the arithmetic series formula:
S_n = n/2 * (a_1 + a_n),
where S_n represents the sum of the first n terms, n is the number of terms (number of rows), a_1 is the first term, and a_n is the nth term.
In this case, the sum of the number of seats in all the rows is:
S_n = n/2 * (40 + [40 + (n - 1)3]).
With these steps, you can find the number of seats in each row and the total number of seats in the auditorium.
To find the total number of seats in the auditorium, you can use the concept of arithmetic series and the explicit formula.
First, let's find the number of seats in each row using the explicit formula. The explicit formula for an arithmetic sequence is:
An = A1 + (n - 1)d
Where An represents the nth term, A1 is the first term, n is the number of terms, and d is the common difference.
In this case, the first row (A1) contains 40 seats, and the common difference (d) between each row is 3 seats. Since there are 49 rows, we can find the number of seats in any row (An) using the formula:
An = 40 + (n - 1) * 3
Now, to find the total number of seats, we will use the sum formula for an arithmetic series:
Sn = (n/2) * (A1 + An)
where Sn represents the sum of the series, n is the number of terms, A1 is the first term, and An is the last term.
Since we have 49 rows, we substitute the values into the formula:
Sn = (49/2) * (40 + (49 - 1) * 3)
Simplifying this expression will give us the total number of seats in the auditorium.