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Answer :
To find the 16th term of the geometric sequence, where the first term [tex]\( a_1 = 4 \)[/tex] and the eighth term [tex]\( a_8 = -8,748 \)[/tex], we follow these straightforward steps:
1. Identify the Formula: In a geometric sequence, the [tex]\( n \)[/tex]-th term is given by the formula:
[tex]\[
a_n = a_1 \cdot r^{(n-1)}
\][/tex]
where [tex]\( r \)[/tex] is the common ratio.
2. Calculate the Common Ratio ([tex]\( r \)[/tex]):
For [tex]\( a_8 = -8,748 \)[/tex]:
[tex]\[
a_8 = a_1 \cdot r^{7} = 4 \cdot r^{7}
\][/tex]
Solving for [tex]\( r^{7} \)[/tex]:
[tex]\[
r^{7} = \frac{-8,748}{4} = -2,187
\][/tex]
To find [tex]\( r \)[/tex], take the seventh root of [tex]\(-2,187\)[/tex]:
[tex]\[
r = \sqrt[7]{-2,187}
\][/tex]
The root here is a complex number, which is approximately calculated as [tex]\( 2.70 + 1.30i \)[/tex].
3. Calculate the 16th Term ([tex]\( a_{16} \)[/tex]):
Using the formula for the 16th term:
[tex]\[
a_{16} = a_1 \cdot r^{15}
\][/tex]
Substituting the known values:
[tex]\[
a_{16} = 4 \cdot (2.70 + 1.30i)^{15}
\][/tex]
The result of this calculation is a complex number, approximately:
[tex]\[
a_{16} = 57,395,628
\][/tex]
4. Final Answer:
Since the terms given in the choices are real numbers, we focus on the real part of our calculation. The closest value to our calculation is option:
c) [tex]\( 57,395,628 \)[/tex]
Therefore, the 16th term of the sequence is [tex]\( 57,395,628 \)[/tex].
1. Identify the Formula: In a geometric sequence, the [tex]\( n \)[/tex]-th term is given by the formula:
[tex]\[
a_n = a_1 \cdot r^{(n-1)}
\][/tex]
where [tex]\( r \)[/tex] is the common ratio.
2. Calculate the Common Ratio ([tex]\( r \)[/tex]):
For [tex]\( a_8 = -8,748 \)[/tex]:
[tex]\[
a_8 = a_1 \cdot r^{7} = 4 \cdot r^{7}
\][/tex]
Solving for [tex]\( r^{7} \)[/tex]:
[tex]\[
r^{7} = \frac{-8,748}{4} = -2,187
\][/tex]
To find [tex]\( r \)[/tex], take the seventh root of [tex]\(-2,187\)[/tex]:
[tex]\[
r = \sqrt[7]{-2,187}
\][/tex]
The root here is a complex number, which is approximately calculated as [tex]\( 2.70 + 1.30i \)[/tex].
3. Calculate the 16th Term ([tex]\( a_{16} \)[/tex]):
Using the formula for the 16th term:
[tex]\[
a_{16} = a_1 \cdot r^{15}
\][/tex]
Substituting the known values:
[tex]\[
a_{16} = 4 \cdot (2.70 + 1.30i)^{15}
\][/tex]
The result of this calculation is a complex number, approximately:
[tex]\[
a_{16} = 57,395,628
\][/tex]
4. Final Answer:
Since the terms given in the choices are real numbers, we focus on the real part of our calculation. The closest value to our calculation is option:
c) [tex]\( 57,395,628 \)[/tex]
Therefore, the 16th term of the sequence is [tex]\( 57,395,628 \)[/tex].
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