To find the $n$-th term of an arithmetic sequence, we use the formula:

$a_n = a + (n - 1)d$

where:

• $a$ is the first term of the sequence,
• $d$ is the common difference, and
• $n$ is the term number.

For the sequence $2, 4, 6, 8, 10, \ldots$:

• The first term $a = 2$.
• The common difference $d = 4 - 2 = 2$.

Substitute these values into the formula:

$a_n = 2 + (n - 1) \cdot 2$

Simplify this expression:

$a_n = 2 + 2n - 2 = 2n$

So, the $n$-th term of the sequence is:

$a_n = 2n$

Summary

For the sequence $2, 4, 6, 8, 10, \ldots$, the $n$-th term is given by: $a_n = 2n$

Would you like further details or have any questions?

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Here are five related questions to extend understanding:

1. What is the sum of the first $n$ terms of this arithmetic sequence?
2. How would you find the common difference if it were not given?
3. Can you derive the formula for the $n$-th term of an arithmetic sequence in general?
4. What is the 15th term of this sequence?
5. How would the formula change if the common difference were negative?

Tip: In an arithmetic sequence, the common difference $d$ can be found by subtracting any term from the term that follows it.