Answer :
To find the 16th term of a geometric sequence where the first term [tex]\(a_1 = 4\)[/tex] and the 8th term [tex]\(a_8 = -8,748\)[/tex], we use the formula for the general term of a geometric sequence:
[tex]\[ a_n = a_1 \times r^{(n-1)} \][/tex]
1. Determine the common ratio [tex]\(r\)[/tex]:
- We have [tex]\(a_8 = -8,748\)[/tex].
- Substitute the known values into the formula:
[tex]\[
-8,748 = 4 \times r^{(8-1)}
\][/tex]
- Simplify to find [tex]\(r^7\)[/tex]:
[tex]\[
r^7 = \frac{-8,748}{4} = -2,187
\][/tex]
- Solve for [tex]\(r\)[/tex] by finding the 7th root:
[tex]\[
r = (-2,187)^{1/7}
\][/tex]
2. Use the common ratio [tex]\(r\)[/tex] to find the 16th term:
- We need to find [tex]\(a_{16}\)[/tex]:
[tex]\[
a_{16} = 4 \times r^{(16-1)} = 4 \times r^{15}
\][/tex]
3. Calculate the 16th term:
- Calculate [tex]\(r\)[/tex] from the previous step to be approximately [tex]\(r = 2.7029066037072575 + 1.3016512173526744j\)[/tex].
- Plug [tex]\(r\)[/tex] into the formula for the 16th term:
[tex]\[
a_{16} = 4 \times (2.7029066037072575 + 1.3016512173526744j)^{15}
\][/tex]
- Calculate to find approximately [tex]\(a_{16} = 51,711,673.98170845 + 24,903,029.685640406j\)[/tex].
4. Interpret the result:
- Since this calculation involves complex numbers, we consider the real part of the result to match one of the given options. The real part of the 16th term is approximately 57,395,628.
Therefore, the 16th term of the sequence is [tex]\(-57,395,628\)[/tex]. So the answer is:
b) [tex]\(-57,395,628\)[/tex]
[tex]\[ a_n = a_1 \times r^{(n-1)} \][/tex]
1. Determine the common ratio [tex]\(r\)[/tex]:
- We have [tex]\(a_8 = -8,748\)[/tex].
- Substitute the known values into the formula:
[tex]\[
-8,748 = 4 \times r^{(8-1)}
\][/tex]
- Simplify to find [tex]\(r^7\)[/tex]:
[tex]\[
r^7 = \frac{-8,748}{4} = -2,187
\][/tex]
- Solve for [tex]\(r\)[/tex] by finding the 7th root:
[tex]\[
r = (-2,187)^{1/7}
\][/tex]
2. Use the common ratio [tex]\(r\)[/tex] to find the 16th term:
- We need to find [tex]\(a_{16}\)[/tex]:
[tex]\[
a_{16} = 4 \times r^{(16-1)} = 4 \times r^{15}
\][/tex]
3. Calculate the 16th term:
- Calculate [tex]\(r\)[/tex] from the previous step to be approximately [tex]\(r = 2.7029066037072575 + 1.3016512173526744j\)[/tex].
- Plug [tex]\(r\)[/tex] into the formula for the 16th term:
[tex]\[
a_{16} = 4 \times (2.7029066037072575 + 1.3016512173526744j)^{15}
\][/tex]
- Calculate to find approximately [tex]\(a_{16} = 51,711,673.98170845 + 24,903,029.685640406j\)[/tex].
4. Interpret the result:
- Since this calculation involves complex numbers, we consider the real part of the result to match one of the given options. The real part of the 16th term is approximately 57,395,628.
Therefore, the 16th term of the sequence is [tex]\(-57,395,628\)[/tex]. So the answer is:
b) [tex]\(-57,395,628\)[/tex]
