There are 21 rows of seats in an amphitheatre. Each row has 4 seats more than the row in front. The last row has 100 seats. How many seats are there altogether in the amphitheatre
this is just an A.P. with
d = 4
a = 100-20*4 = 20
So, the sum of 21 rows is
S = 21/2 (20+100)
To find the total number of seats in the amphitheatre, we can use the formula for the sum of an arithmetic series.
The formula for the sum of an arithmetic series is given by: Sn = (n/2) * (a + l), where Sn is the sum of the series, n is the number of terms, a is the first term, and l is the last term.
Let's find the number of terms (n) first:
Since the last row has 100 seats, and each row has 4 seats more than the row in front, we can work backwards to find the number of terms.
The second-to-last row has 100 - 4 = 96 seats.
The third-to-last row has 96 - 4 = 92 seats.
Continuing this pattern, we can find the first term (a) and the number of terms (n).
a = 100 - (21 - 1) * 4 = 100 - 20 * 4 = 20 seats
n = 21
Now let's substitute the values into the formula:
Sn = (n/2) * (a + l)
= (21/2) * (20 + 100)
= 10.5 * 120
= 1260
Therefore, there are 1260 seats altogether in the amphitheatre.
To find the total number of seats in the amphitheatre, we need to determine the number of seats in each row and then sum them up.
Let's first find out how many seats are in the first row. We know that the last row has 100 seats, and each row has 4 seats more than the row in front. So, the first row will have 100 - 4*(number of rows before the last row) seats.
The number of rows before the last row is obtained by subtracting 1 from the total number of rows since the last row is included in the total. In this case, there are 21 rows in total, so the number of rows before the last row is 21 - 1 = 20.
Now we can calculate the number of seats in the first row:
Number of seats in the first row = 100 - 4*(number of rows before the last row)
= 100 - 4*20
= 100 - 80
= 20
Since each row has 4 seats more than the row in front, we can find the number of seats in each subsequent row by adding 4 to the previous row's seat count:
Number of seats in the second row = 20 + 4 = 24
Number of seats in the third row = 24 + 4 = 28
Number of seats in the fourth row = 28 + 4 = 32
...
and so on until the last row with 100 seats.
Now, to find the total number of seats in the amphitheatre, we add up the number of seats in each row:
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 = 20 + 24 + 28 + 32 + ... + 100
We can use the arithmetic series formula to calculate the sum of the seats:
Total number of seats = (number of terms / 2) * (first term + last term)
In this case, the first term is 20, the last term is 100, and the number of terms is the total number of rows (21).
Total number of seats = (21 / 2) * (20 + 100)
= 10.5 * 120
= 1260
Therefore, there are 1260 seats altogether in the amphitheatre.