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

User Bimo
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1 Answer

4 votes

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)

User Sajjad Pourali
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