To find the 57th term in the sequence [tex]\(13, 8, 3, -2, -7, -12, \ldots \)[/tex], we follow these steps:
1. Identify the first term ([tex]\( a \)[/tex]) and the common difference ([tex]\( d \)[/tex]):
- The first term ([tex]\( a \)[/tex]) of the sequence is [tex]\( 13 \)[/tex].
- To find the common difference ([tex]\( d \)[/tex]), subtract the first term from the second term:
[tex]\[
d = 8 - 13 = -5
\][/tex]
2. Use the formula for the [tex]\( n \)[/tex]-th term of an arithmetic sequence:
The [tex]\( n \)[/tex]-th term of an arithmetic sequence can be found using the formula:
[tex]\[
a_n = a + (n - 1) \cdot d
\][/tex]
3. Substitute the known values into the formula:
- [tex]\( a = 13 \)[/tex]
- [tex]\( d = -5 \)[/tex]
- [tex]\( n = 57 \)[/tex]
Substituting these values in, we get:
[tex]\[
a_{57} = 13 + (57 - 1) \cdot (-5)
\][/tex]
4. Calculate the [tex]\( 57 \)[/tex]-th term:
First, calculate the term within the parentheses:
[tex]\[
57 - 1 = 56
\][/tex]
Next, multiply this result by the common difference [tex]\( d \)[/tex]:
[tex]\[
56 \cdot (-5) = -280
\][/tex]
Finally, add this result to the first term [tex]\( a \)[/tex]:
[tex]\[
a_{57} = 13 + (-280) = 13 - 280 = -267
\][/tex]
Therefore, the 57th term of the given arithmetic sequence is [tex]\( \mathbf{-267} \)[/tex].
