Question: Write an equation to describe the sequence below, and then find the 5th term. Use $n$ to represent the position of a term in the sequence, where $n = 1$ for the first term.
-3, -12, -48, ...
Write your answer using decimals and integers.
aₙ = [ ]([ ])^(n - 1)
a₅ = [ ]
1. **State the problem:** We are given the sequence $$-3, -12, -48, ...$$ and need to find the formula for the $n$th term $a_n$ and then calculate the 5th term $a_5$.
2. **Identify the type of sequence:** This is a geometric sequence because each term is obtained by multiplying the previous term by a constant ratio.
3. **Find the common ratio $r$:**
$$r = \frac{a_2}{a_1} = \frac{-12}{-3} = 4$$
4. **Write the general formula for a geometric sequence:**
$$a_n = a_1 \times r^{n-1}$$
where $a_1$ is the first term and $r$ is the common ratio.
5. **Substitute the known values:**
$$a_n = -3 \times 4^{n-1}$$
6. **Calculate the 5th term $a_5$:**
$$a_5 = -3 \times 4^{5-1} = -3 \times 4^4 = -3 \times 256 = -768$$
**Final answers:**
$$a_n = -3(4)^{n-1}$$
$$a_5 = -768$$