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Answer :
To identify the 16th term of the geometric sequence, follow these steps:
1. Understand the Problem:
- We are given a geometric sequence with the first term [tex]\( a_1 = 4 \)[/tex].
- The eighth term [tex]\( a_8 = -8/40 \)[/tex], which simplifies to [tex]\( a_8 = -0.2 \)[/tex].
- We need to find the 16th term of this sequence.
2. Formula for General Term:
- The formula for the [tex]\( n \)[/tex]-th term of a geometric sequence is:
[tex]\[
a_n = a_1 \times r^{n-1}
\][/tex]
where [tex]\( r \)[/tex] is the common ratio.
3. Calculate the Common Ratio:
- We know [tex]\( a_8 = a_1 \times r^{7} \)[/tex].
- Substitute the known values:
[tex]\[
-0.2 = 4 \times r^{7}
\][/tex]
- Solve for [tex]\( r^{7} \)[/tex]:
[tex]\[
r^{7} = \frac{-0.2}{4} = -0.05
\][/tex]
- To find [tex]\( r \)[/tex], take the seventh root:
[tex]\[
r = (-0.05)^{1/7}
\][/tex]
- This will give a complex value because we are taking an odd root of a negative number.
4. Find the 16th Term:
- Use the formula for the 16th term:
[tex]\[
a_{16} = 4 \times r^{15}
\][/tex]
- Since [tex]\( r \)[/tex] is a complex number, [tex]\( r^{15} \)[/tex] will also be a complex number.
- Multiply by the first term to obtain the complex 16th term.
5. Interpret the Result:
- The calculation shows that the 16th term is approximately a small complex number.
The true values for [tex]\( r \)[/tex] and [tex]\( a_{16} \)[/tex] are complex, but the real part or modulus of [tex]\( a_{16} \)[/tex] is exceedingly small, which explains why it doesn't match any large integer numbers given in the multiple-choice options.
The closest approach to understanding this solution in the context of the provided options is recognizing that it is not feasible to assign a particular real number for the 16th term, as the output involves complex numbers due to the nature of calculations with negative powers.
1. Understand the Problem:
- We are given a geometric sequence with the first term [tex]\( a_1 = 4 \)[/tex].
- The eighth term [tex]\( a_8 = -8/40 \)[/tex], which simplifies to [tex]\( a_8 = -0.2 \)[/tex].
- We need to find the 16th term of this sequence.
2. Formula for General Term:
- The formula for the [tex]\( n \)[/tex]-th term of a geometric sequence is:
[tex]\[
a_n = a_1 \times r^{n-1}
\][/tex]
where [tex]\( r \)[/tex] is the common ratio.
3. Calculate the Common Ratio:
- We know [tex]\( a_8 = a_1 \times r^{7} \)[/tex].
- Substitute the known values:
[tex]\[
-0.2 = 4 \times r^{7}
\][/tex]
- Solve for [tex]\( r^{7} \)[/tex]:
[tex]\[
r^{7} = \frac{-0.2}{4} = -0.05
\][/tex]
- To find [tex]\( r \)[/tex], take the seventh root:
[tex]\[
r = (-0.05)^{1/7}
\][/tex]
- This will give a complex value because we are taking an odd root of a negative number.
4. Find the 16th Term:
- Use the formula for the 16th term:
[tex]\[
a_{16} = 4 \times r^{15}
\][/tex]
- Since [tex]\( r \)[/tex] is a complex number, [tex]\( r^{15} \)[/tex] will also be a complex number.
- Multiply by the first term to obtain the complex 16th term.
5. Interpret the Result:
- The calculation shows that the 16th term is approximately a small complex number.
The true values for [tex]\( r \)[/tex] and [tex]\( a_{16} \)[/tex] are complex, but the real part or modulus of [tex]\( a_{16} \)[/tex] is exceedingly small, which explains why it doesn't match any large integer numbers given in the multiple-choice options.
The closest approach to understanding this solution in the context of the provided options is recognizing that it is not feasible to assign a particular real number for the 16th term, as the output involves complex numbers due to the nature of calculations with negative powers.
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