Answer:
There are a total of 2720 seats in the theater.
The sum of the first 12 positive even integers is 156.
Step-by-step explanation:
(a) To find the total number of seats in the theater with 20 rows, we can use the arithmetic series formula:
Sₙ = n/2 ×(a₁+aₙ)
where:
Sₙ is the sum of the series,
n is the number of terms in the series (number of rows),
a₁ is the first term of the series (number of seats in the first row),
aₙ is the last term of the series (number of seats in the last row).
Given:
n=20 (number of rows),
a₁ =60 (number of seats in the first row),
The pattern is increasing by 8 seats each row.
We need to find a, the number of seats in the 20th row.
a₂₀=a₁ +(n−1)×d
a₂₀=60+(20−1)×8
a₂₀=60+19×8
a₂₀ =60+152
a₂₀ =212
Now, we can calculate the sum of the series:
S₂₀= 20/2 ×(60+212)
S₂₀=10×272
S₂₀ =2720
So, there are a total of 2720 seats in the theater.
(b) The sum of the first 12 positive even integers can be found using the formula for the sum of an arithmetic series:
Sₙ = n/2 ×(a₁ + aₙ)
where:
Sₙ is the sum of the series,
n is the number of terms in the series,
a₁ is the first term of the series,
aₙ is the last term of the series.
Given:
n=12 (number of terms),
a₁ =2 (first even integer),
a₁₂ =2×12=24 (last even integer).
Using the formula:
S₁₂ =12/2 ×(2+24)
S₁₂ =6×26
S₁₂ =156
So, the sum of the first 12 positive even integers is 156.
