Question

Write a recursive sequence that represents the sequence defined by the following explicit formula: an = = -4(-2)^n

Answer 1

In order to find the sequence represented by the given explicit

formula we have to plug in the values of n.

The formula is

$a_n=-4(-2)^n$

hence, for n=1, we have

$a_1=-4(-2)^1$

which is equal to

$\begin{gathered} a_1=-4(-2)^{} \\ a_1=8 \end{gathered}$

In the same way, for n=2, we have

$\begin{gathered} a_2=-4(-2)^2 \\ a_2=-4(4) \\ a_2=-16 \end{gathered}$

For n=3, it yields

$\begin{gathered} a_3=-4(-2)^3 \\ a_3=-4(-8) \\ a_3=32 \end{gathered}$

For n=4, we obtain

$\begin{gathered} a_4=-4(-2)^4 \\ a_4=-4(16) \\ a_4=-64 \end{gathered}$

and so on.

Now,

$\begin{gathered} a_{n-1}=-4(-2)^{n-1} \\ a_{n-1}=-4(-2)^n(-2)^{-1} \\ a_{n-1}=\frac{-4(-2^{})^n}{-2} \\ -2\cdot a_{n-1}=-4(-2)^n \end{gathered}$

since, we know that

$\begin{gathered} a_n=-4(-2)^n \\ we\text{ have } \\ -2\cdot a_{n-1}=a_n \end{gathered}$

in other words, we have that

$a_n=-2\cdot a_{n-1}$