Question Write an explicit formula for $$ a_{n} $$, the $$ n^{ ext {th }} $$ term of the sequence $$ 180,30,5, \ldots $$ Answer Attempt 1 out of 3 $$ a_{n}=\square $$ Submit Answer

Question

Question Write an explicit formula for $$ a_{n} $$, the $$ n^{	ext {th }} $$ term of the sequence $$ 180,30,5, \ldots $$ Answer Attempt 1 out of 3 $$ a_{n}=\square $$ Submit Answer

UpStudy AI · Accepted Answer

To find an explicit formula for the $$ n^{\text{th}} $$ term of the sequence $$ 180, 30, 5, \ldots $$, we first need to identify the pattern in the sequence.

1. **Identify the terms:**
   - $$ a_1 = 180 $$
   - $$ a_2 = 30 $$
   - $$ a_3 = 5 $$

2. **Calculate the ratios between consecutive terms:**
   - $$ \frac{a_2}{a_1} = \frac{30}{180} = \frac{1}{6} $$
   - $$ \frac{a_3}{a_2} = \frac{5}{30} = \frac{1}{6} $$

The ratio between consecutive terms is constant, indicating that this is a geometric sequence.

3. **Determine the common ratio:**
   - The common ratio $$ r = \frac{1}{6} $$.

4. **Use the formula for the $$ n^{\text{th}} $$ term of a geometric sequence:**
   The formula for the $$ n^{\text{th}} $$ term of a geometric sequence is given by:
   $$
   a_n = a_1 \cdot r^{n-1}
   $$
   where $$ a_1 $$ is the first term and $$ r $$ is the common ratio.

5. **Substituting the known values:**
   - $$ a_1 = 180 $$
   - $$ r = \frac{1}{6} $$

Thus, the explicit formula for the $$ n^{\text{th}} $$ term is:
$$
a_n = 180 \cdot \left(\frac{1}{6}\right)^{n-1}
$$

Now, I will submit this formula as the final answer.

The explicit formula for $$ a_n $$ is:
$$
a_n = 180 \cdot \left(\frac{1}{6}\right)^{n-1}
$$
The explicit formula for $$ a_n $$ is: $$ a_n = 180 \cdot \left(\frac{1}{6}\right)^{n-1} $$

Question Write an explicit formula for \( a_{n} \), the \( n^{\text {th }} \) term of the sequence \( 180,30,5, \ldots \) Answer Attempt 1 out of 3 \( a_{n}=\square \) Submit Answer

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