Answer :
Certainly! 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], we first need to determine the common ratio of the sequence.
### Step 1: Calculate the Common Ratio
The formula for the nth term of a geometric sequence is:
[tex]\[
a_n = a_1 \cdot r^{(n-1)}
\][/tex]
For the eighth term:
[tex]\[
a_8 = a_1 \cdot r^{7}
\][/tex]
Substitute the given values:
[tex]\[
-8,748 = 4 \cdot r^{7}
\][/tex]
Solve for [tex]\( r^7 \)[/tex]:
[tex]\[
r^{7} = \frac{-8,748}{4} = -2,187
\][/tex]
Now, we need to find the actual common ratio, [tex]\( r \)[/tex]:
[tex]\[
r = (-2,187)^{\frac{1}{7}}
\][/tex]
### Step 2: Use the Common Ratio to Find the 16th Term
Once you have the common ratio [tex]\( r \)[/tex], use it to determine the 16th term:
[tex]\[
a_{16} = a_1 \cdot r^{15}
\][/tex]
Substitute [tex]\( a_1 = 4 \)[/tex] and the calculated [tex]\( r \)[/tex]:
After performing these calculations, the 16th term is approximately:
[tex]\[
a_{16} = 57,395,628
\][/tex]
Therefore, the correct answer is:
c [tex]\( \quad 57,395,628 \)[/tex]
### Step 1: Calculate the Common Ratio
The formula for the nth term of a geometric sequence is:
[tex]\[
a_n = a_1 \cdot r^{(n-1)}
\][/tex]
For the eighth term:
[tex]\[
a_8 = a_1 \cdot r^{7}
\][/tex]
Substitute the given values:
[tex]\[
-8,748 = 4 \cdot r^{7}
\][/tex]
Solve for [tex]\( r^7 \)[/tex]:
[tex]\[
r^{7} = \frac{-8,748}{4} = -2,187
\][/tex]
Now, we need to find the actual common ratio, [tex]\( r \)[/tex]:
[tex]\[
r = (-2,187)^{\frac{1}{7}}
\][/tex]
### Step 2: Use the Common Ratio to Find the 16th Term
Once you have the common ratio [tex]\( r \)[/tex], use it to determine the 16th term:
[tex]\[
a_{16} = a_1 \cdot r^{15}
\][/tex]
Substitute [tex]\( a_1 = 4 \)[/tex] and the calculated [tex]\( r \)[/tex]:
After performing these calculations, the 16th term is approximately:
[tex]\[
a_{16} = 57,395,628
\][/tex]
Therefore, the correct answer is:
c [tex]\( \quad 57,395,628 \)[/tex]
