Find the Term-to-Term Rule: Analyzing Sequence 2, 5, 8

To solve for the term-to-term rule of the sequence $2, 5, 8, \ldots$, follow these steps:

Step 1: Identify the First Term and Common Difference
The first term $a_1$ of the sequence is $2$.
To find the common difference $d$, subtract the first term from the second term: $5 - 2 = 3$.
Thus, the common difference $d$ is $3$.

Step 2: Derive the Formula for the $n$-th Term
Since the sequence is arithmetic, use the general formula for an arithmetic sequence:
$a_n = a_1 + (n-1) \cdot d$
Substitute the known values, $a_1 = 2$ and $d = 3$:
$a_n = 2 + (n-1) \cdot 3$
Simplify the expression:
$a_n = 2 + 3n - 3$
Combine like terms:
$a_n = 3n - 1$

Step 3: Verify the Formula
Check the derived formula $a_n = 3n - 1$ with the terms given in the sequence:
- For $n = 1$, $a_1 = 3 \times 1 - 1 = 2$ (matches the first term).
- For $n = 2$, $a_2 = 3 \times 2 - 1 = 5$ (matches the second term).
- For $n = 3$, $a_3 = 3 \times 3 - 1 = 8$ (matches the third term).

Therefore, the term-to-term rule for the sequence is $a_n = 3n - 1$.