If an = 3" + n3", n = 0, 1, 2, ..., then an satisfies

(A) an = 3an-1+9an-2

(B) an = 3an-1-9an-2

(C) an = 6an-1 +9an-2

(D) an = 6an-1-9an-2

Given the sequence \( a_n = 3^n + n \cdot 3^n \) for \( n = 0, 1, 2, \ldots \), we want to determine which recurrence relation it satisfies.

Let's first expand the given formula for \( a_n \) and observe the behavior for some values of \( n \):

1. **Expression for \( a_n \):**
   \[
   a_n = 3^n + n \cdot 3^n
   \]

2. **Expression for \( a_{n-1} \):**
   \[
   a_{n-1} = 3^{n-1} + (n-1) \cdot 3^{n-1}
   \]

3. **Expression for \( a_{n-2} \):**
   \[
   a_{n-2} = 3^{n-2} + (n-2) \cdot 3^{n-2}
   \]

We want to find a relation involving \( a_n \), \( a_{n-1} \), and \( a_{n-2} \). Let's express \( a_n \) in terms of \( a_{n-1} \) and \( a_{n-2} \).

### Step-by-step Calculation

Let's manipulate these expressions:

\[
a_n = 3^n + n \cdot 3^n = (1 + n)3^n
\]

Multiplying \( a_{n-1} \) by 3:

\[
3a_{n-1} = 3 \left(3^{n-1} + (n-1) \cdot 3^{n-1}\right) = 3^n + (n-1) \cdot 3^n = (1 + n - 1)3^n = n3^n
\]

Now, \( 9a_{n-2} \):

\[
9a_{n-2} = 9 \left(3^{n-2} + (n-2) \cdot 3^{n-2}\right) = 3^n + (n-2) \cdot 3^n
\]

Now let's combine these results to express \( a_n \):

\[
a_n = 3a_{n-1} + 9a_{n-2}
\]

Thus, the correct recurrence relation is:

\[
a_n = 3a_{n-1} + 9a_{n-2}
\]

### Correct Answer:

The correct option is **(A) \( a_n = 3a_{n-1} + 9a_{n-2} \)**.

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Would you like more details on the steps involved in deriving this solution, or do you have any other questions?

Here are some related questions that might interest you:

1. What is the method for finding recurrence relations in sequences?
2. How can you determine the closed-form solution for a given recurrence relation?
3. What is the characteristic equation of a recurrence relation, and how is it used?
4. Can you solve non-homogeneous recurrence relations? If so, how?
5. How do you find the initial conditions necessary to uniquely determine a sequence defined by a recurrence relation?

**Tip:** When working with sequences defined by recurrence relations, always check the base cases and ensure the recurrence fits all known terms.

Explore how to derive a recurrence relation for a sequence defined by an = 3^n + n * 3^n. Step-by-step solution provided with detailed calculations and explanation. Suitable for high school students studying sequences and recurrence relations.

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