Question: Write an equation to describe the sequence below. Use $n$ to represent the position of a term in the sequence, where $n = 1$ for the first term.
$4, 12, 36, ...$
Write your answer using decimals and integers.
aₙ = [ ] ( [ ] )^{n - 1}
1. **State the problem:** We are given the sequence $4, 12, 36, ...$ and need to find the equation for the $n$th term $a_n$ of this sequence, where $n$ is the position of the term starting at $n=1$.
2. **Identify the type of sequence:** The sequence appears to be geometric because each term is multiplied by the same number to get the next term.
3. **Formula for geometric sequence:** The $n$th term of a geometric sequence is given by:
$$a_n = a_1 \times r^{n-1}$$
where $a_1$ is the first term and $r$ is the common ratio.
4. **Find the first term $a_1$:** The first term is given as $4$.
5. **Find the common ratio $r$:** Divide the second term by the first term:
$$r = \frac{12}{4} = 3$$
Check with the third term:
$$\frac{36}{12} = 3$$
So the common ratio is $3$.
6. **Write the explicit formula:** Substitute $a_1 = 4$ and $r = 3$ into the formula:
$$a_n = 4 \times 3^{n-1}$$
7. **Final answer:**
$$a_n = 4(3)^{n-1}$$
This formula describes the sequence $4, 12, 36, ...$ where $n$ is the term position starting at $1$.