the first row in a theater has 8 seats. each subsequent row has 3 more seats than the row before it. there are 10 rows in the orchestra section. how many seats are in the orchestra section?
each time you move to the next row you should add 3 to the row before it.
8+11+14+17+20+23+26+29+32+35=215 seats
To find the total number of seats in the orchestra section, we need to calculate the number of seats in each row and then sum them up.
The first row has 8 seats. Each subsequent row has 3 more seats than the row before it.
To find the number of seats in each row, we can use the arithmetic sequence formula:
an = a1 + (n - 1)d
Where:
an is the value of the nth term
a1 is the first term
n is the number of terms
d is the common difference
In this case, a1 = 8 (seats in the first row), n = 10 (number of rows), and d = 3 (increase in seats for each subsequent row).
Using the formula, we can find the number of seats in each row:
a1 = 8 (seats in the first row)
a2 = a1 + (2 - 1)d = 8 + (1)(3) = 11 (seats in the second row)
a3 = a1 + (3 - 1)d = 8 + (2)(3) = 14 (seats in the third row)
...
a10 = a1 + (10 - 1)d = 8 + (9)(3) = 8 + 27 = 35 (seats in the tenth row)
Now, we can sum up the number of seats in each row:
Total seats in the orchestra section = a1 + a2 + a3 + ... + a10
Total seats = 8 + 11 + 14 + ... + 35
To calculate this sum, we can use the arithmetic series formula:
Sn = (n/2)(a1 + an)
Where:
Sn is the sum of the arithmetic sequence
n is the number of terms
a1 is the first term
an is the last term
Using the formula, we can find the sum of the seats:
Total seats = (10/2)(8 + 35)
= 5(43)
= 215
Therefore, there are 215 seats in the orchestra section.