To determine the explicit rule for the sequence given:

$n = 1, 2, 3 \quad \text{and} \quad a_n = 4, 8, 12$

Step 1: Analyze the sequence

• The sequence increases by 4 each time: $8 - 4 = 4 \quad \text{and} \quad 12 - 8 = 4$ This tells us the sequence has a common difference of 4, which indicates it's an arithmetic sequence.

Step 2: General form of an arithmetic sequence

The general formula for an arithmetic sequence is: $a_n = a_1 + (n - 1) \cdot d$ Where:

• $a_1$ = the first term ($4$),
• $d$ = the common difference ($4$).

Step 3: Substitute into the formula

Substituting $a_1 = 4$ and $d = 4$: $a_n = 4 + (n - 1) \cdot 4$ Simplify: $a_n = 4 + 4n - 4$ $a_n = 4n$

Step 4: Verify the formula

We test the formula $a_n = 4n$ with the given values:

• For $n = 1$, $a_1 = 4 \cdot 1 = 4$ ✔
• For $n = 2$, $a_2 = 4 \cdot 2 = 8$ ✔
• For $n = 3$, $a_3 = 4 \cdot 3 = 12$ ✔

The formula works for all given terms.

Final Answer:

The explicit rule for the sequence is: $\boxed{a_n = 4n}$ This corresponds to option b.

Do you need further clarification or more details? Let me know!

Related Questions:

1. What is the formula for the $n$-th term of an arithmetic sequence?
2. How do you find the common difference in a sequence?
3. How do you check if a sequence is arithmetic?
4. What is the sum of the first $n$-terms of this sequence?
5. What changes in the formula if the sequence starts at a different number?

Tip: Always verify your formula by plugging in the given terms. It ensures accuracy!