Answer :
To find the 16th term of the geometric sequence where the first term [tex]\(a_1 = 4\)[/tex] and the 8th term [tex]\(a_8 = -8,748\)[/tex], follow these steps:
1. Identify the formula for the nth term of a geometric sequence:
The formula is:
[tex]\[
a_n = a_1 \cdot r^{(n-1)}
\][/tex]
where [tex]\(a_n\)[/tex] is the nth term, [tex]\(a_1\)[/tex] is the first term, [tex]\(r\)[/tex] is the common ratio, and [tex]\(n\)[/tex] is the term number.
2. Relate the given terms to the formula:
You're given [tex]\(a_1 = 4\)[/tex] and [tex]\(a_8 = -8,748\)[/tex]. Plug these values into the formula for the 8th term:
[tex]\[
a_8 = a_1 \cdot r^{(8-1)}
\][/tex]
[tex]\[
-8,748 = 4 \cdot r^7
\][/tex]
3. Solve for the common ratio [tex]\(r\)[/tex]:
[tex]\[
r^7 = \frac{-8,748}{4}
\][/tex]
[tex]\[
r^7 = -2,187
\][/tex]
Calculate [tex]\(r\)[/tex] by taking the 7th root of -2,187.
4. Find the 16th term:
Using the formula:
[tex]\[
a_{16} = a_1 \cdot r^{(16-1)}
\][/tex]
[tex]\[
a_{16} = 4 \cdot r^{15}
\][/tex]
5. Calculate [tex]\(a_{16}\)[/tex]:
With the value of [tex]\(r\)[/tex] already determined, plug it in to find:
[tex]\[
a_{16} = 4 \cdot (r^{15})
\][/tex]
After performing these calculations, you end up with:
- The common ratio [tex]\(r\)[/tex] is approximately [tex]\((2.7029066037072575+1.3016512173526744j)\)[/tex].
- The 16th term [tex]\(a_{16}\)[/tex] is approximately [tex]\((57,395,628)\)[/tex].
Thus, the correct answer is [tex]\(57,395,628\)[/tex], which corresponds to option c.
1. Identify the formula for the nth term of a geometric sequence:
The formula is:
[tex]\[
a_n = a_1 \cdot r^{(n-1)}
\][/tex]
where [tex]\(a_n\)[/tex] is the nth term, [tex]\(a_1\)[/tex] is the first term, [tex]\(r\)[/tex] is the common ratio, and [tex]\(n\)[/tex] is the term number.
2. Relate the given terms to the formula:
You're given [tex]\(a_1 = 4\)[/tex] and [tex]\(a_8 = -8,748\)[/tex]. Plug these values into the formula for the 8th term:
[tex]\[
a_8 = a_1 \cdot r^{(8-1)}
\][/tex]
[tex]\[
-8,748 = 4 \cdot r^7
\][/tex]
3. Solve for the common ratio [tex]\(r\)[/tex]:
[tex]\[
r^7 = \frac{-8,748}{4}
\][/tex]
[tex]\[
r^7 = -2,187
\][/tex]
Calculate [tex]\(r\)[/tex] by taking the 7th root of -2,187.
4. Find the 16th term:
Using the formula:
[tex]\[
a_{16} = a_1 \cdot r^{(16-1)}
\][/tex]
[tex]\[
a_{16} = 4 \cdot r^{15}
\][/tex]
5. Calculate [tex]\(a_{16}\)[/tex]:
With the value of [tex]\(r\)[/tex] already determined, plug it in to find:
[tex]\[
a_{16} = 4 \cdot (r^{15})
\][/tex]
After performing these calculations, you end up with:
- The common ratio [tex]\(r\)[/tex] is approximately [tex]\((2.7029066037072575+1.3016512173526744j)\)[/tex].
- The 16th term [tex]\(a_{16}\)[/tex] is approximately [tex]\((57,395,628)\)[/tex].
Thus, the correct answer is [tex]\(57,395,628\)[/tex], which corresponds to option c.
