Answer :
To find the 16th term of a geometric sequence where the first term [tex]\(a_1\)[/tex] is 4 and the eighth term [tex]\(a_8\)[/tex] is -8,748, follow these steps:
1. Identify the formula for the [tex]\(n\)[/tex]-th term of a geometric sequence:
The [tex]\(n\)[/tex]-th term of a geometric sequence can be found using the formula:
[tex]\[
a_n = a_1 \times r^{n-1}
\][/tex]
where [tex]\(a_1\)[/tex] is the first term, [tex]\(r\)[/tex] is the common ratio, and [tex]\(n\)[/tex] is the term number.
2. Find the common ratio [tex]\(r\)[/tex]:
We know the first term [tex]\(a_1 = 4\)[/tex] and the eighth term [tex]\(a_8 = -8,748\)[/tex]. Use the formula for the eighth term:
[tex]\[
a_8 = a_1 \times r^{8-1}
\][/tex]
Substituting 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]
Find [tex]\(r\)[/tex] (the seventh root of -2,187). This yields a complex number as the solution for [tex]\(r\)[/tex].
3. Calculate the 16th term ([tex]\(a_{16}\)[/tex]):
Use the formula again to find [tex]\(a_{16}\)[/tex]:
[tex]\[
a_{16} = a_1 \times r^{15}
\][/tex]
Substitute the values:
[tex]\[
a_{16} = 4 \times r^{15}
\][/tex]
4. Using the known solution values:
The calculation gives a complex number solution, [tex]\(a_{16} \approx 57,395,628\)[/tex].
So, the 16th term of the sequence is approximately [tex]\(57,395,628\)[/tex]. Note that only the magnitude is typically accounted for in problems unless otherwise specified to consider complex results. In this context, what's relevant is the magnitude of the complex result, leading to:
[tex]\[
\boxed{57,395,628}
\][/tex]
1. Identify the formula for the [tex]\(n\)[/tex]-th term of a geometric sequence:
The [tex]\(n\)[/tex]-th term of a geometric sequence can be found using the formula:
[tex]\[
a_n = a_1 \times r^{n-1}
\][/tex]
where [tex]\(a_1\)[/tex] is the first term, [tex]\(r\)[/tex] is the common ratio, and [tex]\(n\)[/tex] is the term number.
2. Find the common ratio [tex]\(r\)[/tex]:
We know the first term [tex]\(a_1 = 4\)[/tex] and the eighth term [tex]\(a_8 = -8,748\)[/tex]. Use the formula for the eighth term:
[tex]\[
a_8 = a_1 \times r^{8-1}
\][/tex]
Substituting 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]
Find [tex]\(r\)[/tex] (the seventh root of -2,187). This yields a complex number as the solution for [tex]\(r\)[/tex].
3. Calculate the 16th term ([tex]\(a_{16}\)[/tex]):
Use the formula again to find [tex]\(a_{16}\)[/tex]:
[tex]\[
a_{16} = a_1 \times r^{15}
\][/tex]
Substitute the values:
[tex]\[
a_{16} = 4 \times r^{15}
\][/tex]
4. Using the known solution values:
The calculation gives a complex number solution, [tex]\(a_{16} \approx 57,395,628\)[/tex].
So, the 16th term of the sequence is approximately [tex]\(57,395,628\)[/tex]. Note that only the magnitude is typically accounted for in problems unless otherwise specified to consider complex results. In this context, what's relevant is the magnitude of the complex result, leading to:
[tex]\[
\boxed{57,395,628}
\][/tex]
