Certainly! Let's solve the problem step-by-step. We will find the 57th term of the given sequence: 13, 8, 3, -2, -7, -12, ...
To determine the 57th term in this sequence, we need to identify the first term and the common difference, and then use the formula for the nth term of an arithmetic sequence.
### Step-by-Step Solution
1. Identify the first term (a):
The first term of the sequence is given as 13.
[tex]\[
a = 13
\][/tex]
2. Calculate the common difference (d):
The difference between any two consecutive terms in the sequence can be calculated by subtracting the first term from the second term. Hence,
[tex]\[
d = 8 - 13 = -5
\][/tex]
3. Formula for the nth term of an arithmetic sequence:
The general formula to find the nth term ([tex]\(a_n\)[/tex]) of an arithmetic sequence is:
[tex]\[
a_n = a + (n - 1) \times d
\][/tex]
Here, we need to find the 57th term ([tex]\(a_{57}\)[/tex]). So, [tex]\(n = 57\)[/tex].
4. Substitute the values into the formula:
[tex]\[
a_{57} = 13 + (57 - 1) \times (-5)
\][/tex]
5. Calculate inside the parenthesis first:
[tex]\[
a_{57} = 13 + 56 \times (-5)
\][/tex]
6. Multiply 56 by -5:
[tex]\[
56 \times (-5) = -280
\][/tex]
7. Add the result to the first term:
[tex]\[
a_{57} = 13 + (-280)
\][/tex]
[tex]\[
a_{57} = 13 - 280
\][/tex]
[tex]\[
a_{57} = -267
\][/tex]
### Conclusion
The 57th term of the sequence is [tex]\(-267\)[/tex].
