Question Write an explicit formula for $$ a_{n} $$, the $$ n^{ ext {th }} $$ term of the sequence $$ 8,-4,2, \ldots $$ 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 $$ 8,-4,2, \ldots $$ Answer Attempt 1 out of 3 $$ a_{n}=\square $$ Submit Answer

UpStudy AI · Accepted Answer

To find the explicit formula for the $$ n^{\text{th}} $$ term of the sequence $$ 8, -4, 2, \ldots $$, we first need to identify the pattern in the sequence.

1. **Identify the first term**: 
   - The first term $$ a_1 = 8 $$.

2. **Identify the second term**: 
   - The second term $$ a_2 = -4 $$.

3. **Identify the third term**: 
   - The third term $$ a_3 = 2 $$.

4. **Calculate the differences**:
   - The difference between the first and second term: $$ -4 - 8 = -12 $$.
   - The difference between the second and third term: $$ 2 - (-4) = 6 $$.

5. **Identify the pattern**:
   - The sequence appears to be decreasing and then increasing, suggesting a linear relationship.

6. **Finding the common ratio**:
   - The ratio between the first and second term: $$ \frac{-4}{8} = -\frac{1}{2} $$.
   - The ratio between the second and third term: $$ \frac{2}{-4} = -\frac{1}{2} $$.

This indicates that the sequence is geometric with a common ratio of $$ -\frac{1}{2} $$.

7. **General formula for a geometric sequence**:
   - The general 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.

8. **Substituting the values**:
   - Here, $$ a_1 = 8 $$ and $$ r = -\frac{1}{2} $$.
   - Therefore, the explicit formula for the $$ n^{\text{th}} $$ term is:
     $$
     a_n = 8 \left(-\frac{1}{2}\right)^{(n-1)}
     $$

Now, I will submit the final answer for $$ a_n $$.

The explicit formula for the $$ n^{\text{th}} $$ term of the sequence is:
$$
a_n = 8 \left(-\frac{1}{2}\right)^{(n-1)}
$$
The explicit formula for the $$ n^{\text{th}} $$ term of the sequence is: $$ a_n = 8 \left(-\frac{1}{2}\right)^{(n-1)} $$

Question Write an explicit formula for \( a_{n} \), the \( n^{\text {th }} \) term of the sequence \( 8,-4,2, \ldots \) Answer Attempt 1 out of 3 \( a_{n}=\square \) Submit Answer

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