To solve for the equation describing the given sequence $3, 9, 27, \dots$, we notice that this sequence represents a geometric sequence because each term is multiplied by the same factor to get the next.

Step 1: Identify the common ratio $r$

To determine $r$, divide the second term by the first term: $r = \frac{9}{3} = 3$ The common ratio is $3$.

Step 2: General form of a geometric sequence

The formula for a geometric sequence is: $a_n = a_1 \cdot r^{n-1}$ where:

• $a_1$ is the first term (3),
• $r$ is the common ratio (3),
• $n$ is the position of the term.

Step 3: Write the equation

Substitute $a_1 = 3$ and $r = 3$ into the formula: $a_n = 3 \cdot 3^{n-1}$

Final Answer:

The equation to describe the sequence is: $a_n = 3(3)^{n-1}$

Let me know if you'd like more explanations or examples!

Related Questions:

1. How do you find the $n$-th term of a geometric sequence?
2. What is the general form of an arithmetic sequence?
3. How can you determine the common ratio in a geometric sequence?
4. What happens if the common ratio is a fraction in a geometric sequence?
5. How do you calculate the sum of the first $n$-terms in a geometric series?

Tip: In geometric sequences, multiplying or dividing each term by a constant gives the next term!