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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 :

We start by using the formula for the \( n \)th term of a geometric sequence:

$$
a_n = a_1 \cdot r^{n-1}.
$$

1. Given that \( a_1 = 4 \) and \( a_8 = -8748 \), we have:

$$
a_8 = 4 \cdot r^{7} = -8748.
$$

2. Solve for \( r^7 \):

$$
r^{7} = \frac{-8748}{4} = -2187.
$$

3. Notice that

$$
-2187 = -3^7,
$$

so it follows that

$$
r = -3.
$$

4. To find the 16th term, use the formula with \( n = 16 \):

$$
a_{16} = 4 \cdot (-3)^{15}.
$$

5. Since

$$
(-3)^{15} = -3^{15} = -14348907,
$$

we have:

$$
a_{16} = 4 \cdot \left(-14348907\right) = -57395628.
$$

Thus, the 16th term is

$$
\boxed{-57395628}.
$$

This corresponds to option b.

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