Answer :
To determine the 33rd term of the given arithmetic sequence [tex]\( 12, 7, 2, -3, -8, \ldots \)[/tex], we can use the formula for the [tex]\( n \)[/tex]-th term of an arithmetic sequence. The formula is:
[tex]\[ a_n = a_1 + (n-1) \cdot d \][/tex]
Where:
- [tex]\( a_n \)[/tex] is the [tex]\( n \)[/tex]-th term,
- [tex]\( a_1 \)[/tex] is the first term,
- [tex]\( n \)[/tex] is the term number,
- [tex]\( d \)[/tex] is the common difference.
### Step-by-Step Solution:
1. Identify the first term ([tex]\( a_1 \)[/tex]):
The first term of the sequence is [tex]\( a_1 = 12 \)[/tex].
2. Determine the common difference ([tex]\( d \)[/tex]):
The common difference is the difference between consecutive terms. Calculating the difference between the first and second terms:
[tex]\[ d = 7 - 12 = -5 \][/tex]
3. Identify the term number ([tex]\( n \)[/tex]):
We are asked to find the 33rd term, so [tex]\( n = 33 \)[/tex].
4. Plug these values into the [tex]\( n \)[/tex]-th term formula:
[tex]\[ a_{33} = 12 + (33-1) \cdot (-5) \][/tex]
Simplifying inside the parentheses:
[tex]\[ a_{33} = 12 + 32 \cdot (-5) \][/tex]
Now, multiply:
[tex]\[ a_{33} = 12 + (-160) \][/tex]
Combine the terms:
[tex]\[ a_{33} = 12 - 160 \][/tex]
[tex]\[ a_{33} = -148 \][/tex]
### Conclusion:
The 33rd term of the given arithmetic sequence is [tex]\(-148\)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{-148} \][/tex]
[tex]\[ a_n = a_1 + (n-1) \cdot d \][/tex]
Where:
- [tex]\( a_n \)[/tex] is the [tex]\( n \)[/tex]-th term,
- [tex]\( a_1 \)[/tex] is the first term,
- [tex]\( n \)[/tex] is the term number,
- [tex]\( d \)[/tex] is the common difference.
### Step-by-Step Solution:
1. Identify the first term ([tex]\( a_1 \)[/tex]):
The first term of the sequence is [tex]\( a_1 = 12 \)[/tex].
2. Determine the common difference ([tex]\( d \)[/tex]):
The common difference is the difference between consecutive terms. Calculating the difference between the first and second terms:
[tex]\[ d = 7 - 12 = -5 \][/tex]
3. Identify the term number ([tex]\( n \)[/tex]):
We are asked to find the 33rd term, so [tex]\( n = 33 \)[/tex].
4. Plug these values into the [tex]\( n \)[/tex]-th term formula:
[tex]\[ a_{33} = 12 + (33-1) \cdot (-5) \][/tex]
Simplifying inside the parentheses:
[tex]\[ a_{33} = 12 + 32 \cdot (-5) \][/tex]
Now, multiply:
[tex]\[ a_{33} = 12 + (-160) \][/tex]
Combine the terms:
[tex]\[ a_{33} = 12 - 160 \][/tex]
[tex]\[ a_{33} = -148 \][/tex]
### Conclusion:
The 33rd term of the given arithmetic sequence is [tex]\(-148\)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{-148} \][/tex]
