Let's solve the given problem step-by-step.
### Part a) Write a rule for the number of seats in the nth row.
We are given that the number of seats in each row follows a specific pattern:
- The first row has 10 seats.
- Each subsequent row has 1 seat more than the previous one.
This forms an arithmetic sequence where:
- The first term [tex]\( a_1 \)[/tex] is 10.
- The common difference [tex]\( d \)[/tex] is 1.
The formula for the nth term of an arithmetic sequence is given by:
[tex]\[ a_n = a_1 + (n - 1) d \][/tex]
Here:
- [tex]\( a_1 = 10 \)[/tex]
- [tex]\( d = 1 \)[/tex]
Therefore, the rule for the number of seats in the nth row ([tex]\( a_n \)[/tex]) is:
[tex]\[ a_n = 10 + (n - 1) \cdot 1 \][/tex]
[tex]\[ a_n = 10 + n - 1 \][/tex]
[tex]\[ a_n = 9 + n \][/tex]
So, the rule can be expressed as:
[tex]\[ a_n = 9 + n \][/tex]
### Part b) Find the total number of seats in the auditorium with 25 rows.
To find the total number of seats, we need to sum the number of seats in each row from the 1st row to the 25th row. This requires summing the arithmetic sequence from [tex]\( n = 1 \)[/tex] to [tex]\( n = 25 \)[/tex].
The sum [tex]\( S_n \)[/tex] of the first [tex]\( n \)[/tex] terms of an arithmetic sequence can be calculated using the formula:
[tex]\[ S_n = \frac{n}{2} \left( 2a_1 + (n - 1) d \right) \][/tex]
Here:
- [tex]\( n = 25 \)[/tex]
- [tex]\( a_1 = 10 \)[/tex]
- [tex]\( d = 1 \)[/tex]
Plugging in these values:
[tex]\[ S_{25} = \frac{25}{2} \left( 2 \cdot 10 + (25 - 1) \cdot 1 \right) \][/tex]
[tex]\[ S_{25} = \frac{25}{2} \left( 20 + 24 \right) \][/tex]
[tex]\[ S_{25} = \frac{25}{2} \times 44 \][/tex]
[tex]\[ S_{25} = \frac{25 \times 44}{2} \][/tex]
[tex]\[ S_{25} = \frac{1100}{2} \][/tex]
[tex]\[ S_{25} = 550 \][/tex]
Therefore, the total number of seats in the auditorium is [tex]\( 550 \)[/tex].
