Certainly! Let's break down each part of the problem step-by-step.
### a. Write an explicit rule for the number of seats that are in the nth row of the theater.
The problem states that the first row has 12 seats, and each subsequent row has 4 more seats than the previous row. This suggests an arithmetic sequence where:
- The first term (a₁) is 12.
- The common difference (d) is 4.
The formula for the nth term of an arithmetic sequence is given by:
[tex]\[ a_n = a_1 + (n - 1) \cdot d \][/tex]
So, for the nth row:
[tex]\[ \text{seats\_in\_nth\_row} = 12 + (n - 1) \cdot 4 \][/tex]
This gives us the explicit rule for the number of seats in the nth row.
### b. How many seats are in the 10th row?
To find the number of seats in the 10th row, we substitute [tex]\( n = 10 \)[/tex] into the formula derived above:
[tex]\[ \text{seats\_in\_10th\_row} = 12 + (10 - 1) \cdot 4 \][/tex]
Calculating this:
[tex]\[ \text{seats\_in\_10th\_row} = 12 + 9 \cdot 4 \][/tex]
[tex]\[ \text{seats\_in\_10th\_row} = 12 + 36 \][/tex]
[tex]\[ \text{seats\_in\_10th\_row} = 48 \][/tex]
Therefore, there are 48 seats in the 10th row.
### c. How many seats are there total in all 43 rows?
To find the total number of seats in all 43 rows, we need to sum the terms of our arithmetic sequence from the 1st row to the 43rd row.
The formula to find the sum of the first n terms (Sₙ) of an arithmetic sequence is:
[tex]\[ S_n = \frac{n}{2} \cdot (2a_1 + (n - 1)d) \][/tex]
Substituting [tex]\( n = 43 \)[/tex], [tex]\( a_1 = 12 \)[/tex], and [tex]\( d = 4 \)[/tex] into the formula:
[tex]\[ S_{43} = \frac{43}{2} \cdot (2 \cdot 12 + (43 - 1) \cdot 4) \][/tex]
Let's calculate the sum step-by-step:
[tex]\[ S_{43} = \frac{43}{2} \cdot (24 + 42 \cdot 4) \][/tex]
[tex]\[ S_{43} = \frac{43}{2} \cdot (24 + 168) \][/tex]
[tex]\[ S_{43} = \frac{43}{2} \cdot 192 \][/tex]
[tex]\[ S_{43} = 43 \cdot 96 \][/tex]
Thus:
[tex]\[ S_{43} = 4128 \][/tex]
Therefore, the total number of seats in all 43 rows is 4128.
