Question

Which equation is an explicit formula that represents the sequence?

Answer 1

In general, a geometric sequence is given by the formula.

$a_n=ar^{n-1}$

Then, in our case,

$\begin{gathered} a_1=-4 \\ \text{and} \\ a_1=ar^{1-1}=ar^0=a \\ \Rightarrow a=-4_{} \end{gathered}$

And, since the second term is 12,

$\begin{gathered} a_2=12 \\ \Rightarrow12=-4r^{2-1}=-4r \\ \Rightarrow r=-3 \end{gathered}$

Thus, the formula is

$\Rightarrow a_n=-4(-3)^{n-1}$

However, there is a mistake in the sequence given in the question.

Notice that if n=4, none of the options gives us a_4=144.

$\begin{gathered} a_4=-4(3)^{4-1}=-4(3)^3=-108 \\ a_4=-4(12)^{4-1}=-4(12)^3=-6912 \\ a_4=-4(-3)^{4-1}=-4(-3)^3=108 \\ a_4=-3(-4)^{4-1}=-3(-4)^3=192 \end{gathered}$

The third option gives us correctly the first three terms of the sequence. The answer is the third option (top to bottom)