Find the Pattern Rule: Analyzing Sequence 3,15,27,39,41 Using Variable n

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

• Step 1: Recognize the pattern in the series.
• Step 2: Identify the first term and the common difference.
• Step 3: Derive the formula for the sequence.

Now, let's work through each step:

Step 1: The given series is $3, 15, 27, 39, 41$. Observing the first few terms, $3, 15, 27, 39$, we note each subsequent number increases by $12$, forming an arithmetic sequence. The number $41$ does not follow this arithmetic sequence pattern.

Step 2: The first term $a_1$ is $3$, and the common difference $d$ is $12$, as derived from verifying the difference between each two successive terms.

Step 3: We use the arithmetic sequence formula:
$\displaystyle a(n) = a_1 + (n-1) \times d$

Substitute the known values:
$\displaystyle a(n) = 3 + (n-1) \times 12$

This formula describes the arithmetic sequence for the original numbers $3, 15, 27, 39$ but not for $41$.

Therefore, the solution to the problem is:

$a(n) = 3 + (n-1) \times 12$.