Determine the Arithmetic Sequence Rule: What's the Pattern for 2, 10, 18?

What is the term-to-term rule of the following sequence?

2, 10, 18, ...

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

• Step 1: Identify the first term $a_1$ of the sequence.
• Step 2: Determine the common difference $d$.
• Step 3: Use the formula for the $n$-th term of an arithmetic sequence.

Let's work through each step:

Step 1: The first term $a_1$ of the sequence is 2.

Step 2: The common difference $d$ is calculated by subtracting the first term from the second term: $10 - 2 = 8$.

Step 3: Apply the formula for the $n$-th term of an arithmetic sequence:
The formula is $a_n = a_1 + (n-1) \cdot d$.
Substitute $a_1 = 2$ and $d = 8$ into the formula:

$a_n = 2 + (n-1) \cdot 8$

Simplify this expression:
$a_n = 2 + 8n - 8 = 8n - 6$

Therefore, the term-to-term rule for the sequence is $8n - 6$.