Answer :
To solve this problem, we are looking for the 16th term of a geometric sequence given that the first term is [tex]\(a_1 = 4\)[/tex] and the eighth term is [tex]\(a_8 = -8,748\)[/tex].
Here's how you can find it step-by-step:
1. Understand the formula for the nth term of a geometric sequence:
The formula is:
[tex]\[
a_n = a_1 \times r^{(n-1)}
\][/tex]
where [tex]\(r\)[/tex] is the common ratio, and [tex]\(n\)[/tex] is the term number in the sequence.
2. Find the common ratio [tex]\(r\)[/tex]:
You have the 8th term:
[tex]\[
a_8 = a_1 \times r^7
\][/tex]
Substitute the known values:
[tex]\[
-8,748 = 4 \times r^7
\][/tex]
Solve for [tex]\(r^7\)[/tex]:
[tex]\[
r^7 = \frac{-8,748}{4} = -2,187
\][/tex]
Now solve for [tex]\(r\)[/tex]:
[tex]\[
r = (-2,187)^{1/7}
\][/tex]
3. Find the 16th term [tex]\(a_{16}\)[/tex]:
Use the nth-term formula again:
[tex]\[
a_{16} = a_1 \times r^{15}
\][/tex]
Substitute the first term and the power of the ratio:
[tex]\[
a_{16} = 4 \times r^{15}
\][/tex]
4. Calculate the result:
The calculations show that:
[tex]\[
a_{16} \approx 57,395,628
\][/tex]
Therefore, the 16th term of the geometric sequence is approximately [tex]\(57,395,628\)[/tex], which matches one of the given choices. So, the correct answer is [tex]\(57,395,628\)[/tex].
Here's how you can find it step-by-step:
1. Understand the formula for the nth term of a geometric sequence:
The formula is:
[tex]\[
a_n = a_1 \times r^{(n-1)}
\][/tex]
where [tex]\(r\)[/tex] is the common ratio, and [tex]\(n\)[/tex] is the term number in the sequence.
2. Find the common ratio [tex]\(r\)[/tex]:
You have the 8th term:
[tex]\[
a_8 = a_1 \times r^7
\][/tex]
Substitute the known values:
[tex]\[
-8,748 = 4 \times r^7
\][/tex]
Solve for [tex]\(r^7\)[/tex]:
[tex]\[
r^7 = \frac{-8,748}{4} = -2,187
\][/tex]
Now solve for [tex]\(r\)[/tex]:
[tex]\[
r = (-2,187)^{1/7}
\][/tex]
3. Find the 16th term [tex]\(a_{16}\)[/tex]:
Use the nth-term formula again:
[tex]\[
a_{16} = a_1 \times r^{15}
\][/tex]
Substitute the first term and the power of the ratio:
[tex]\[
a_{16} = 4 \times r^{15}
\][/tex]
4. Calculate the result:
The calculations show that:
[tex]\[
a_{16} \approx 57,395,628
\][/tex]
Therefore, the 16th term of the geometric sequence is approximately [tex]\(57,395,628\)[/tex], which matches one of the given choices. So, the correct answer is [tex]\(57,395,628\)[/tex].
