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Identify the 16th term of a geometric sequence where [tex]a_1 = 4[/tex] and [tex]a_8 = -8,748[/tex].

A. [tex]-172,186,884[/tex]
B. [tex]-57,395,628[/tex]
C. [tex]57,395,628[/tex]
D. [tex]172,186,884[/tex]

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]

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