We are given the arithmetic sequence:

$2, -3, -8, \dots$

In an arithmetic sequence, the difference between consecutive terms is constant. This difference is called the common difference, denoted as $d$.

Step 1: Find the common difference $d$

To find $d$, subtract the first term from the second term:

$d = -3 - 2 = -5$

Thus, the common difference $d = -5$.

Step 2: Use the formula for the $n$-th term of an arithmetic sequence

The formula for the $n$-th term of an arithmetic sequence is:

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

Where:

• $a_n$ is the $n$-th term,
• $a_1$ is the first term,
• $d$ is the common difference, and
• $n$ is the term number.

Step 3: Find the 64th term

We are asked to find the 64th term, so set $n = 64$, $a_1 = 2$, and $d = -5$:

$a_{64} = 2 + (64 - 1) \cdot (-5)$ $a_{64} = 2 + 63 \cdot (-5)$ $a_{64} = 2 - 315$ $a_{64} = -313$

Final Answer:

The 64th term of the sequence is $\boxed{-313}$.

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Would you like further details on how I arrived at this solution, or do you have any other questions?

Here are some follow-up questions:

1. How would the formula change if the sequence was a geometric sequence instead of an arithmetic one?
2. How can you find the sum of the first $n$ terms of an arithmetic sequence?
3. How can you use the common difference to predict future terms in the sequence?
4. What happens if the common difference is positive instead of negative?
5. Can you explain the concept of the "first term" in more detail?

Tip: When identifying the common difference in an arithmetic sequence, always subtract the first term from the second term (or any two consecutive terms) to find $d$.