Sure! Let's solve the problem step by step.
### Step 1: Understanding the given information and the formula
We are given:
- [tex]\(a_1 = 16\)[/tex]
- [tex]\(a_5 = 15006\)[/tex]
We need to find the 17th term, [tex]\(a_{17}\)[/tex], of the geometric sequence.
The general form of the [tex]\(n\)[/tex]-th term of a geometric sequence is given by:
[tex]\[ a_n = a_1 \cdot r^{(n-1)} \][/tex]
where:
- [tex]\(a_n\)[/tex] is the [tex]\(n\)[/tex]-th term,
- [tex]\(a_1\)[/tex] is the first term,
- [tex]\(r\)[/tex] is the common ratio,
- [tex]\(n\)[/tex] is the term number.
### Step 2: Determine the common ratio [tex]\(r\)[/tex]
We know:
[tex]\[ a_5 = 15006 \][/tex]
Using the formula for the 5th term:
[tex]\[ a_5 = a_1 \cdot r^{(5-1)} \][/tex]
[tex]\[ 15006 = 16 \cdot r^4 \][/tex]
To find [tex]\(r\)[/tex], we divide 15006 by 16:
[tex]\[ r^4 = \frac{15006}{16} \][/tex]
Next, we take the fourth root of [tex]\(\frac{15006}{16}\)[/tex]:
[tex]\[ r = \left( \frac{15006}{16} \right)^{\frac{1}{4}} \][/tex]
### Step 3: Calculate the common ratio [tex]\(r\)[/tex]
After calculating,
[tex]\[ r \approx 5.53 \][/tex]
(rounded to the nearest hundredth)
### Step 4: Determine the 17th term [tex]\(a_{17}\)[/tex]
Using the formula for the 17th term:
[tex]\[ a_{17} = a_1 \cdot r^{(17-1)} \][/tex]
[tex]\[ a_{17} = 16 \cdot 5.53^{16} \][/tex]
### Step 5: Calculate the 17th term [tex]\(a_{17}\)[/tex]
After calculating,
[tex]\[ a_{17} \approx 12379406399648.74 \][/tex]
(rounded to the nearest hundredth)
### Summary
Thus, the common ratio [tex]\(r\)[/tex] is approximately [tex]\(5.53\)[/tex] and the 17th term [tex]\(a_{17}\)[/tex] is approximately [tex]\(12379406399648.74\)[/tex].
