To solve for $a_9$ given that $a_3 = 18$ and $a_6 = 486$, we can assume that this sequence is either an arithmetic sequence or a geometric sequence. Since the values grow significantly from $a_3$ to $a_6$, this suggests it is likely a geometric sequence.

In a geometric sequence, each term is given by: $a_n = a_1 \cdot r^{n-1}$ where $a_1$ is the first term and $r$ is the common ratio.

Step 1: Set up equations for $a_3$ and $a_6$

1. For $a_3$: $a_3 = a_1 \cdot r^{3-1} = a_1 \cdot r^2 = 18$

2. For $a_6$: $a_6 = a_1 \cdot r^{6-1} = a_1 \cdot r^5 = 486$

Step 2: Divide the equations to eliminate $a_1$

Dividing the second equation by the first gives: $\frac{a_6}{a_3} = \frac{a_1 \cdot r^5}{a_1 \cdot r^2} = r^3$ $\frac{486}{18} = r^3$ $27 = r^3$ $r = 3$

Step 3: Substitute $r$ back to find $a_1$

Using $a_3 = a_1 \cdot r^2$: $18 = a_1 \cdot 3^2$ $18 = a_1 \cdot 9$ $a_1 = 2$

Step 4: Find $a_9$

Now, we can find $a_9$ using the formula for the $n$-th term: $a_9 = a_1 \cdot r^{9-1} = 2 \cdot 3^8$

Calculating $3^8$: $3^8 = 6561$ $a_9 = 2 \cdot 6561 = 13122$

Final Answer

$a_9 = 13122$

Would you like further details on any step?

Here are 5 related questions that expand on this concept:

1. What would $a_4$ and $a_7$ be in this sequence?
2. How would you determine if a sequence is geometric or arithmetic if only given two terms?
3. Can you find the sum of the first 10 terms of this sequence?
4. How does the formula for the sum of a geometric series differ from that of an arithmetic series?
5. What happens to the common ratio if the terms of a geometric sequence decrease instead of increase?

Tip: When solving problems involving sequences, always check if the terms grow by addition (arithmetic) or by multiplication (geometric) to identify the type of sequence.