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Answer :
To find the 16th term of a geometric sequence where it is given that the first term [tex]\( a_1 = 4 \)[/tex] and the 16th term [tex]\( a_{16} = -8,748 \)[/tex], we need to find out the formula for a geometric sequence and apply it to find the common ratio and the 16th term.
1. Understand the Formula:
The [tex]\( n \)[/tex]-th term of a geometric sequence is given by the formula:
[tex]\[
a_n = a_1 \times r^{(n-1)}
\][/tex]
where [tex]\( a_n \)[/tex] is the [tex]\( n \)[/tex]-th term, [tex]\( a_1 \)[/tex] is the first term, [tex]\( r \)[/tex] is the common ratio, and [tex]\( n \)[/tex] is the number of terms.
2. Apply It to the 16th Term:
For the 16th term, we have:
[tex]\[
a_{16} = a_1 \times r^{15}
\][/tex]
Substituting the given values:
[tex]\[
-8,748 = 4 \times r^{15}
\][/tex]
3. Solve for the Common Ratio ([tex]\( r \)[/tex]):
Divide both sides by 4 to isolate [tex]\( r^{15} \)[/tex]:
[tex]\[
r^{15} = \frac{-8,748}{4} = -2,187
\][/tex]
To find the common ratio [tex]\( r \)[/tex], you calculate the 15th root of -2,187. In a complex number scenario, the computation of the root reveals that [tex]\( r \)[/tex] is a complex number. The calculated approximate values for [tex]\( r \)[/tex] turn out to be:
[tex]\[
r \approx 1.633 + 0.347i
\][/tex]
4. Verification:
Plug the calculated value of [tex]\( r \)[/tex] back into the equation and you would initially get the complex output due to [tex]\( r \)[/tex] being a complex number.
Since the answer choices for the next terms seem to position the result as real numbers, if everything aligns correctly in practice without other requirements for simplifications:
- The calculations inherently consider the complexity involved in powers involving non-integer roots.
Hence, the 16th term is indeed [tex]\(-8,748\)[/tex], and the next steps require the correct application of these formulas and values that help derive other terms if needed based on the sequence's known setup.
1. Understand the Formula:
The [tex]\( n \)[/tex]-th term of a geometric sequence is given by the formula:
[tex]\[
a_n = a_1 \times r^{(n-1)}
\][/tex]
where [tex]\( a_n \)[/tex] is the [tex]\( n \)[/tex]-th term, [tex]\( a_1 \)[/tex] is the first term, [tex]\( r \)[/tex] is the common ratio, and [tex]\( n \)[/tex] is the number of terms.
2. Apply It to the 16th Term:
For the 16th term, we have:
[tex]\[
a_{16} = a_1 \times r^{15}
\][/tex]
Substituting the given values:
[tex]\[
-8,748 = 4 \times r^{15}
\][/tex]
3. Solve for the Common Ratio ([tex]\( r \)[/tex]):
Divide both sides by 4 to isolate [tex]\( r^{15} \)[/tex]:
[tex]\[
r^{15} = \frac{-8,748}{4} = -2,187
\][/tex]
To find the common ratio [tex]\( r \)[/tex], you calculate the 15th root of -2,187. In a complex number scenario, the computation of the root reveals that [tex]\( r \)[/tex] is a complex number. The calculated approximate values for [tex]\( r \)[/tex] turn out to be:
[tex]\[
r \approx 1.633 + 0.347i
\][/tex]
4. Verification:
Plug the calculated value of [tex]\( r \)[/tex] back into the equation and you would initially get the complex output due to [tex]\( r \)[/tex] being a complex number.
Since the answer choices for the next terms seem to position the result as real numbers, if everything aligns correctly in practice without other requirements for simplifications:
- The calculations inherently consider the complexity involved in powers involving non-integer roots.
Hence, the 16th term is indeed [tex]\(-8,748\)[/tex], and the next steps require the correct application of these formulas and values that help derive other terms if needed based on the sequence's known setup.
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