The problem in the image asks for the 57th term of an arithmetic sequence with the initial sequence given as:

$13, 8, 3, -2, -7, -12, \dots$

Step 1: Identify the common difference

The common difference $d$ in an arithmetic sequence is the difference between any two consecutive terms. From the sequence:

$d = 8 - 13 = -5$

Step 2: Use the formula for the nth term of an arithmetic sequence

The formula to find the $n$th term $a_n$ of an arithmetic sequence is:

$a_n = a_1 + (n-1) \times d$

Where:

• $a_1$ is the first term
• $d$ is the common difference
• $n$ is the term number

Step 3: Substitute the values into the formula

Here:

• $a_1 = 13$
• $d = -5$
• $n = 57$

So, the 57th term $a_{57}$ is:

$a_{57} = 13 + (57-1) \times (-5)$

$a_{57} = 13 + 56 \times (-5)$

$a_{57} = 13 - 280$

$a_{57} = -267$

Final Answer:

The 57th term of the sequence is -267.

Would you like more details on this problem, or do you have any other questions?

Here are five related questions to expand your understanding:

1. How do you find the common difference in an arithmetic sequence?
2. What is the formula for the sum of the first $n$ terms of an arithmetic sequence?
3. How can you derive the formula for the $n$th term of an arithmetic sequence?
4. What happens to the terms of an arithmetic sequence when the common difference is positive?
5. How does the common difference affect the nature of the sequence (increasing, decreasing, constant)?

Tip: When dealing with large term numbers in an arithmetic sequence, always ensure you're using the correct values for the first term, common difference, and the term number to avoid calculation errors.