Arithmetic Sequence Rule: Determine the Increment for 4, 14, 24, 34

To solve this problem, we'll follow these steps:

• Step 1: Identify the sequence progression
• Step 2: Find the common difference
• Step 3: Formulate the $n$-th term rule

Now, let's work through each step:

Step 1: Analyze the sequence. The sequence provided is 4, 14, 24, 34, ...

Step 2: Calculate the common difference by subtracting successive terms:
$14 - 4 = 10$, $24 - 14 = 10$, and $34 - 24 = 10$.
The common difference is 10.

Step 3: Use the common difference and the first term to find the formula:
The sequence is arithmetic with the first term $a_1 = 4$ and common difference $d = 10$.
The formula for the $n$-th term of an arithmetic sequence is given by:
$a_n = a_1 + (n-1) \cdot d$.
Plug $a_1 = 4$ and $d = 10$ into the formula:
$a_n = 4 + (n-1) \cdot 10$
$a_n = 4 + 10n - 10$
Thus, $a_n = 10n - 6$.

Therefore, the sequence can be represented by the term-to-term rule:
$10n - 6$.