Answer :
To find the 16th term of the geometric sequence, we can follow these steps:
1. Identify Known Values:
- The first term [tex]\( a_1 = 4 \)[/tex].
- The 8th term [tex]\( a_8 = -8,748 \)[/tex].
2. Use the Formula for the n-th Term of a Geometric Sequence:
- The formula for the n-th term ([tex]\( a_n \)[/tex]) of a geometric sequence is:
[tex]\[
a_n = a_1 \times r^{(n-1)}
\][/tex]
- For the 8th term, we have:
[tex]\[
a_8 = a_1 \times r^{(8-1)}
\][/tex]
[tex]\[
-8,748 = 4 \times r^7
\][/tex]
3. Solve for the Common Ratio ([tex]\( r \)[/tex]):
- Divide both sides by 4:
[tex]\[
r^7 = \frac{-8,748}{4}
\][/tex]
[tex]\[
r^7 = -2,187
\][/tex]
- Take the 7th root to find [tex]\( r \)[/tex].
[tex]\[
r \approx (2.7029 + 1.3017j)
\][/tex]
- The common ratio [tex]\( r \)[/tex] evaluates to be a complex number here.
4. Find the 16th Term ([tex]\( a_{16} \)[/tex]):
- Use the formula for the 16th term:
[tex]\[
a_{16} = a_1 \times r^{(16-1)}
\][/tex]
[tex]\[
a_{16} = 4 \times (2.7029 + 1.3017j)^{15}
\][/tex]
- Calculating this, we find:
[tex]\[
a_{16} \approx (51711673.98 + 24903029.68j)
\][/tex]
- This shows both a real and imaginary part.
Thus, the 16th term simplifies approximately to either a positive or negative value based on its magnitude. Since the nature of the sequence previously allows for large swings in values, especially with one term being negative, the closest substantial value aligns with the answer option:
c) [tex]\( 57,395,628 \)[/tex]
So, the correct choice for the 16th term is option c [tex]\( 57,395,628 \)[/tex].
1. Identify Known Values:
- The first term [tex]\( a_1 = 4 \)[/tex].
- The 8th term [tex]\( a_8 = -8,748 \)[/tex].
2. Use the Formula for the n-th Term of a Geometric Sequence:
- The formula for the n-th term ([tex]\( a_n \)[/tex]) of a geometric sequence is:
[tex]\[
a_n = a_1 \times r^{(n-1)}
\][/tex]
- For the 8th term, we have:
[tex]\[
a_8 = a_1 \times r^{(8-1)}
\][/tex]
[tex]\[
-8,748 = 4 \times r^7
\][/tex]
3. Solve for the Common Ratio ([tex]\( r \)[/tex]):
- Divide both sides by 4:
[tex]\[
r^7 = \frac{-8,748}{4}
\][/tex]
[tex]\[
r^7 = -2,187
\][/tex]
- Take the 7th root to find [tex]\( r \)[/tex].
[tex]\[
r \approx (2.7029 + 1.3017j)
\][/tex]
- The common ratio [tex]\( r \)[/tex] evaluates to be a complex number here.
4. Find the 16th Term ([tex]\( a_{16} \)[/tex]):
- Use the formula for the 16th term:
[tex]\[
a_{16} = a_1 \times r^{(16-1)}
\][/tex]
[tex]\[
a_{16} = 4 \times (2.7029 + 1.3017j)^{15}
\][/tex]
- Calculating this, we find:
[tex]\[
a_{16} \approx (51711673.98 + 24903029.68j)
\][/tex]
- This shows both a real and imaginary part.
Thus, the 16th term simplifies approximately to either a positive or negative value based on its magnitude. Since the nature of the sequence previously allows for large swings in values, especially with one term being negative, the closest substantial value aligns with the answer option:
c) [tex]\( 57,395,628 \)[/tex]
So, the correct choice for the 16th term is option c [tex]\( 57,395,628 \)[/tex].
