To find the 57th term of the given arithmetic sequence, we start by identifying the components of the sequence:
1. The first term ([tex]\(a\)[/tex]) is [tex]\(13\)[/tex].
2. The common difference ([tex]\(d\)[/tex]) is calculated as [tex]\(3 - 8 = -5\)[/tex].
The formula to find the [tex]\(n\)[/tex]th term ([tex]\(a_n\)[/tex]) of an arithmetic sequence is:
[tex]\[ a_n = a + (n - 1) \cdot d \][/tex]
Let's plug in the values:
- [tex]\(a = 13\)[/tex] (the first term)
- [tex]\(d = -5\)[/tex] (the common difference)
- [tex]\(n = 57\)[/tex] (the position of the term we want to find)
Substitute these values into the formula:
[tex]\[ a_{57} = 13 + (57 - 1)\cdot (-5) \][/tex]
Simplify the expression inside the parentheses:
[tex]\[ a_{57} = 13 + 56 \cdot (-5) \][/tex]
Now, multiply [tex]\(56\)[/tex] by [tex]\(-5\)[/tex]:
[tex]\[ 56 \cdot (-5) = -280 \][/tex]
Add this result to the first term:
[tex]\[ a_{57} = 13 + (-280) \][/tex]
This simplifies to:
[tex]\[ a_{57} = 13 - 280 = -267 \][/tex]
So, the common difference is [tex]\(-5\)[/tex], and the 57th term of the sequence is [tex]\(-267\)[/tex].
