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Answer :
To identify the 16th term of a geometric sequence where the first term [tex]\( a_1 = 4 \)[/tex] and the eighth term [tex]\( a_8 = -8,748 \)[/tex], follow these steps:
1. Understand the Formula:
The nth 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 nth term, [tex]\( a_1 \)[/tex] is the first term, and [tex]\( r \)[/tex] is the common ratio.
2. Find the Common Ratio (r):
For the eighth term:
[tex]\[
a_8 = a_1 \times r^{7}
\][/tex]
Substitute the known values:
[tex]\[
-8,748 = 4 \times r^7
\][/tex]
Solve for [tex]\( r^7 \)[/tex]:
[tex]\[
r^7 = \frac{-8,748}{4}
\][/tex]
[tex]\[
r^7 = -2,187
\][/tex]
To find [tex]\( r \)[/tex], take the seventh root of [tex]\(-2,187\)[/tex].
3. Calculate the 16th Term:
For the 16th term:
[tex]\[
a_{16} = a_1 \times r^{15}
\][/tex]
Substitute the known values:
[tex]\[
a_{16} = 4 \times (-2,187)^{\frac{15}{7}}
\][/tex]
4. Determine the Result:
Performing the calculations, the 16th term [tex]\( a_{16} \approx 57,395,628 \)[/tex].
Thus, the 16th term of this geometric sequence is approximately [tex]\( 57,395,628 \)[/tex], which corresponds to option c.
1. Understand the Formula:
The nth 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 nth term, [tex]\( a_1 \)[/tex] is the first term, and [tex]\( r \)[/tex] is the common ratio.
2. Find the Common Ratio (r):
For the eighth term:
[tex]\[
a_8 = a_1 \times r^{7}
\][/tex]
Substitute the known values:
[tex]\[
-8,748 = 4 \times r^7
\][/tex]
Solve for [tex]\( r^7 \)[/tex]:
[tex]\[
r^7 = \frac{-8,748}{4}
\][/tex]
[tex]\[
r^7 = -2,187
\][/tex]
To find [tex]\( r \)[/tex], take the seventh root of [tex]\(-2,187\)[/tex].
3. Calculate the 16th Term:
For the 16th term:
[tex]\[
a_{16} = a_1 \times r^{15}
\][/tex]
Substitute the known values:
[tex]\[
a_{16} = 4 \times (-2,187)^{\frac{15}{7}}
\][/tex]
4. Determine the Result:
Performing the calculations, the 16th term [tex]\( a_{16} \approx 57,395,628 \)[/tex].
Thus, the 16th term of this geometric sequence is approximately [tex]\( 57,395,628 \)[/tex], which corresponds to option c.
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