Answer :
To find the 16th term of the geometric sequence where the first term [tex]\( a_1 \)[/tex] is 4 and the 8th term [tex]\( a_8 \)[/tex] is -8,748, we can follow these steps:
1. Identify the formula for the nth term in a geometric sequence:
The formula for the nth term [tex]\( a_n \)[/tex] of a geometric sequence is given by:
[tex]\[
a_n = a_1 \times r^{(n-1)}
\][/tex]
where [tex]\( a_1 \)[/tex] is the first term and [tex]\( r \)[/tex] is the common ratio.
2. Find the common ratio [tex]\( r \)[/tex]:
We know that the 8th term [tex]\( a_8 = -8,748 \)[/tex]. Plugging this into the formula:
[tex]\[
a_8 = a_1 \times r^{(8-1)} = 4 \times r^{7}
\][/tex]
Solving for [tex]\( r \)[/tex]:
[tex]\[
-8,748 = 4 \times r^7
\][/tex]
[tex]\[
r^7 = \frac{-8,748}{4} = -2,187
\][/tex]
[tex]\[
r = (-2,187)^{1/7}
\][/tex]
3. Calculate the 16th term [tex]\( a_{16} \)[/tex]:
Using the value of [tex]\( r \)[/tex], we find [tex]\( a_{16} \)[/tex] using the formula:
[tex]\[
a_{16} = 4 \times r^{15}
\][/tex]
Using the calculated [tex]\( r \)[/tex], we find:
[tex]\[
a_{16} ≈ 57,395,628
\][/tex]
The 16th term of the geometric sequence is approximately 57,395,628. Therefore, the correct answer is option C.
1. Identify the formula for the nth term in a geometric sequence:
The formula for the nth term [tex]\( a_n \)[/tex] of a geometric sequence is given by:
[tex]\[
a_n = a_1 \times r^{(n-1)}
\][/tex]
where [tex]\( a_1 \)[/tex] is the first term and [tex]\( r \)[/tex] is the common ratio.
2. Find the common ratio [tex]\( r \)[/tex]:
We know that the 8th term [tex]\( a_8 = -8,748 \)[/tex]. Plugging this into the formula:
[tex]\[
a_8 = a_1 \times r^{(8-1)} = 4 \times r^{7}
\][/tex]
Solving for [tex]\( r \)[/tex]:
[tex]\[
-8,748 = 4 \times r^7
\][/tex]
[tex]\[
r^7 = \frac{-8,748}{4} = -2,187
\][/tex]
[tex]\[
r = (-2,187)^{1/7}
\][/tex]
3. Calculate the 16th term [tex]\( a_{16} \)[/tex]:
Using the value of [tex]\( r \)[/tex], we find [tex]\( a_{16} \)[/tex] using the formula:
[tex]\[
a_{16} = 4 \times r^{15}
\][/tex]
Using the calculated [tex]\( r \)[/tex], we find:
[tex]\[
a_{16} ≈ 57,395,628
\][/tex]
The 16th term of the geometric sequence is approximately 57,395,628. Therefore, the correct answer is option C.
