If the first 5 terms of a geometric sequence are $$ \left\{12,6,3, \frac{3}{2}, \frac{3}{4} ight\} $$ then the formula for the $$ n^{ ext {th }} $$ term in the sequence is a.) $$ a_{n}=12 \cdot 2^{n-1} $$ b.) $$ a_{n}=\left(\frac{1}{2} ight) \cdot 12^{n-1} $$ c.) $$ a_{n}=2 \cdot 12^{n-1} $$ d.) $$ a_{n}=12 \cdot\left(\frac{1}{2} ight)^{n-1} $$ (

Question

If the first 5 terms of a geometric sequence are $$ \left\{12,6,3, \frac{3}{2}, \frac{3}{4}ight\} $$ then the formula for the $$ n^{	ext {th }} $$ term in the sequence is a.) $$ a_{n}=12 \cdot 2^{n-1} $$ b.) $$ a_{n}=\left(\frac{1}{2}ight) \cdot 12^{n-1} $$ c.) $$ a_{n}=2 \cdot 12^{n-1} $$ d.) $$ a_{n}=12 \cdot\left(\frac{1}{2}ight)^{n-1} $$ (

UpStudy AI · Accepted Answer

To determine the formula for the $$ n^{\text{th}} $$ term of the given geometric sequence $$ \left\{12, 6, 3, \frac{3}{2}, \frac{3}{4}\right\} $$, follow these steps:

1. **Identify the First Term ($$ a_1 $$) and the Common Ratio ($$ r $$):**

- The first term $$ a_1 $$ is **12**.
   - To find the common ratio $$ r $$, divide the second term by the first term:
     $$
     r = \frac{6}{12} = \frac{1}{2}
     $$
   - Verify the ratio with subsequent terms:
     $$
     \frac{3}{6} = \frac{1}{2}, \quad \frac{\frac{3}{2}}{3} = \frac{1}{2}, \quad \frac{\frac{3}{4}}{\frac{3}{2}} = \frac{1}{2}
     $$
     The common ratio $$ r $$ is consistently $$ \frac{1}{2} $$.

2. **Use the Formula for the $$ n^{\text{th}} $$ Term of a Geometric Sequence:**
   
   The general formula for the $$ n^{\text{th}} $$ term of a geometric sequence is:
   $$
   a_n = a_1 \cdot r^{n-1}
   $$
   
   Substituting the known values:
   $$
   a_n = 12 \cdot \left(\frac{1}{2}\right)^{n-1}
   $$

3. **Compare with the Given Options:**

- **Option d.)** $$ a_n = 12 \cdot \left(\frac{1}{2}\right)^{n-1} $$ matches our derived formula.

**Answer:**  
d.) $$ a_{n}=12 \cdot\left(\frac{1}{2}\right)^{n-1} $$
The formula for the $$ n^{\text{th}} $$ term is $$ a_{n}=12 \cdot\left(\frac{1}{2}\right)^{n-1} $$.

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