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)