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What is the 33rd term of this arithmetic sequence?

[tex]\[ 12, 7, 2, -3, -8, \ldots \][/tex]

A. -158
B. -157
C. -148
D. -147

Sagot :

GinnyAnswer
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]
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