Answer :
To determine the domain for \( n \) in the given arithmetic sequence \( a_n = 4 - 3(n - 1) \), let's break down the problem step by step:
1. Understanding the given sequence:
The formula for the \( n \)-th term of the arithmetic sequence is given by:
[tex]\[ a_n = 4 - 3(n - 1) \][/tex]
Here, \( n \) represents the position of a term in the sequence.
2. Identifying the starting point:
In an arithmetic sequence, \( n \) typically starts from 1 because it represents the position of the terms (first term, second term, etc.).
Substituting \( n = 1 \) into the formula:
[tex]\[ a_1 = 4 - 3(1 - 1) = 4 - 0 = 4 \][/tex]
This confirms that \( a_n \) is correctly defined for \( n = 1 \).
3. Checking the integer property and positivity:
The sequence formula \( 4 - 3(n - 1) \) is defined for all integers \( n \). There is no mathematical restriction on \( n \) other than it representing a term's position, which is usually a positive integer.
4. Determining the domain:
Since \( n \) needs to be an integer starting from 1 (as \( n \) represents the positions like 1st term, 2nd term, etc.), the domain for \( n \) is:
[tex]\[ \text{All integers where } n \geq 1 \][/tex]
5. Listing out the possible options:
- All integers where \( n \geq 1 \)
- All integers where \( n > 1 \)
- All integers where \( n \leq 4 \)
From the above analysis, we see that \( n \) starts from 1 and includes all integers from 1 onward.
Therefore, the correct answer is:
[tex]\[ \text{All integers where } n \geq 1 \][/tex]
1. Understanding the given sequence:
The formula for the \( n \)-th term of the arithmetic sequence is given by:
[tex]\[ a_n = 4 - 3(n - 1) \][/tex]
Here, \( n \) represents the position of a term in the sequence.
2. Identifying the starting point:
In an arithmetic sequence, \( n \) typically starts from 1 because it represents the position of the terms (first term, second term, etc.).
Substituting \( n = 1 \) into the formula:
[tex]\[ a_1 = 4 - 3(1 - 1) = 4 - 0 = 4 \][/tex]
This confirms that \( a_n \) is correctly defined for \( n = 1 \).
3. Checking the integer property and positivity:
The sequence formula \( 4 - 3(n - 1) \) is defined for all integers \( n \). There is no mathematical restriction on \( n \) other than it representing a term's position, which is usually a positive integer.
4. Determining the domain:
Since \( n \) needs to be an integer starting from 1 (as \( n \) represents the positions like 1st term, 2nd term, etc.), the domain for \( n \) is:
[tex]\[ \text{All integers where } n \geq 1 \][/tex]
5. Listing out the possible options:
- All integers where \( n \geq 1 \)
- All integers where \( n > 1 \)
- All integers where \( n \leq 4 \)
From the above analysis, we see that \( n \) starts from 1 and includes all integers from 1 onward.
Therefore, the correct answer is:
[tex]\[ \text{All integers where } n \geq 1 \][/tex]
