Let's address the problem step by step.
Firstly, we observe the given sequence:
[tex]\[ -1, -\frac{1}{3}, -\frac{1}{9}, -\frac{1}{27}, \ldots \][/tex]
This is a geometric sequence because each term after the first is obtained by multiplying the preceding term by a constant ratio.
### Step 1: Determine the Common Ratio
To find the common ratio ([tex]\(r\)[/tex]), we divide the second term by the first term:
[tex]\[ r = \frac{-\frac{1}{3}}{-1} = \frac{1}{3} \][/tex]
So, the common ratio is [tex]\(\frac{1}{3}\)[/tex].
### Step 2: Write the Explicit Formula
The general formula for the [tex]\(n\)[/tex]th term ([tex]\(a_n\)[/tex]) of a geometric sequence is given by:
[tex]\[ a_n = a_1 \cdot r^{(n-1)} \][/tex]
where [tex]\(a_1\)[/tex] is the first term of the sequence and [tex]\(r\)[/tex] is the common ratio.
For the given sequence, [tex]\(a_1 = -1\)[/tex] and [tex]\(r = \frac{1}{3}\)[/tex]. Substituting these values, we get:
[tex]\[ a_n = -1 \cdot \left(\frac{1}{3}\right)^{(n-1)} \][/tex]
#### Explicit Formula:
[tex]\[ a_n = -1 \cdot \left(\frac{1}{3}\right)^{(n-1)} \][/tex]
### Step 3: Find the 10th Term
To find the 10th term of the sequence ([tex]\(a_{10}\)[/tex]), we substitute [tex]\(n = 10\)[/tex] into the explicit formula:
[tex]\[ a_{10} = -1 \cdot \left(\frac{1}{3}\right)^{(10-1)} \][/tex]
[tex]\[ a_{10} = -1 \cdot \left(\frac{1}{3}\right)^9 \][/tex]
Evaluating [tex]\(\left(\frac{1}{3}\right)^9\)[/tex], we get:
[tex]\[ \left(\frac{1}{3}\right)^9 \approx 5.080526342529086 \times 10^{-5} \][/tex]
So, multiplying by [tex]\(-1\)[/tex]:
[tex]\[ a_{10} \approx -1 \cdot 5.080526342529086 \times 10^{-5} \][/tex]
[tex]\[ a_{10} \approx -5.080526342529086 \times 10^{-5} \][/tex]
Thus, the explicit formula for the sequence is:
[tex]\[ a_n = -1 \cdot \left(\frac{1}{3}\right)^{(n-1)} \][/tex]
And the 10th term of the sequence is:
[tex]\[ a_{10} \approx -5.080526342529086 \times 10^{-5} \][/tex]
