Question: 1 — Introduction to Geometric Sequences
LEARNING OBJECTIVE: Determine the formula for a given geometric sequence.
If the first 5 terms of a geometric sequence are ${12, 6, 3, \frac{3}{2}, \frac{3}{4}}$, then the formula for the $n^{th}$ term in the sequence is __________.
a.) $a_n = 12\cdot 2^{n-1}$
b.) $a_n = \frac{1}{2}\cdot 12^{n-1}$
c.) $a_n = 12\cdot \left(\frac{1}{2}\right)^{n-1}$
d.) $a_n = 2\cdot 12^{n-1}$
1. **State the problem:** We are given the first 5 terms of a geometric sequence: $12, 6, 3, \frac{3}{2}, \frac{3}{4}$. We need to find the formula for the $n^{th}$ term, $a_n$.
2. **Recall the formula for the $n^{th}$ term of a geometric sequence:**
$$a_n = a_1 \cdot r^{n-1}$$
where $a_1$ is the first term and $r$ is the common ratio.
3. **Find the common ratio $r$:**
Divide the second term by the first term:
$$r = \frac{6}{12} = \frac{1}{2}$$
Check with the next terms:
$$\frac{3}{6} = \frac{1}{2}, \quad \frac{\frac{3}{2}}{3} = \frac{1}{2}, \quad \frac{\frac{3}{4}}{\frac{3}{2}} = \frac{1}{2}$$
All confirm $r = \frac{1}{2}$.
4. **Write the formula using $a_1 = 12$ and $r = \frac{1}{2}$:**
$$a_n = 12 \cdot \left(\frac{1}{2}\right)^{n-1}$$
5. **Compare with the options:**
Option c) matches exactly.
**Final answer:** $a_n = 12 \cdot \left(\frac{1}{2}\right)^{n-1}$