an auditorium has 50 rows of seats. THe first row has 20 seats, the row has 21 seats, the third row has 22 seats, and so on. each row having one more seat than the previous row. How many seats are there all together?
First row = 20 seats,
second row = 21 seats,
...
nth row = 19+n seats,
...
50th row = 19+50 = 69 seats
Write the number of seats in forward and reverse order:
19 + 69 = 88
20 + 68 = 88
...
44 + 44 = 88
...
69 + 19 = 88
---------
Sum = 50*(88/2) = 2200 seats
21×42
20×40 = 800
Well, it seems like this auditorium has quite a seating arrangement! To find out how many seats there are all together, let's put on our math hats and crunch some numbers.
The first row has 20 seats, the second row has 21 seats, the third row has 22 seats, and so on. We can see that there is a pattern of each row having one more seat than the previous row.
To calculate the total number of seats, we can make use of a little trick involving arithmetic progression. We can use the formula:
n/2 * [2a + (n-1)d],
Where:
n is the number of terms (in this case, the number of rows),
a is the first term (the number of seats in the first row),
and d is the common difference (which is 1 in this case, as there is one more seat in each row).
So, plugging our values into the formula, we get:
50/2 * [2(20) + (50-1)(1)].
Now, let's do some calculations.
50/2 gives us 25,
2(20) gives us 40,
and (50-1)(1) gives us 49.
Finally, if we multiply everything together, we get:
25 * (40 + 49).
And the answer is...
(25 * 89) = 2,225!
So, altogether, there are 2,225 seats in the auditorium. That's a whole lot of room for laughter and applause!
To find the total number of seats in the auditorium, you need to determine the number of seats in each row and add them all together.
In this case, the number of seats in each row follows a pattern where each consecutive row has one more seat than the previous row.
To figure out the number of seats in each row, you can use the formula for an arithmetic sequence:
a_n = a_1 + (n-1)d
where a_n is the nth term of the sequence, a_1 is the first term, n is the number of terms, and d is the common difference.
In this case, the first row (a_1) has 20 seats, and the common difference (d) is 1 because each row has one more seat than the previous row. Therefore, the formula for the number of seats in each row will be:
a_n = 20 + (n-1)
Next, you need to determine the number of rows (n) in the auditorium. It is given that there are 50 rows.
Now, you can calculate the number of seats in each row and add them all together to find the total number of seats in the auditorium:
Total number of seats = a_1 + a_2 + a_3 + ... + a_50
Substituting the values into the formula, you get:
Total number of seats = (20 + (1-1)) + (20 + (2-1)) + (20 + (3-1)) + ... + (20 + (50-1))
Simplifying this equation gives:
Total number of seats = 20 + 21 + 22 + ... + 69
To find the sum of this arithmetic series, you can use the formula:
Sum = (n/2)(a_1 + a_n)
where n is the number of terms and a_n is the nth term.
In this case, n = 50 and a_1 = 20, a_n = 69, so you have:
Sum = (50/2)(20 + 69)
Calculating this, you obtain:
Sum = 25 * 89 = 2225
Therefore, there are a total of 2225 seats in the auditorium.
19+n
u#1=19+1
u#1=20
u#50=19+50
u#1=69
50(20+69)/2