Question

Write a recursive sequence that represents the sequence defined by the following explicit formula: an = 3(-2)n-1

Answer 1

1) Given that for this Sequence we have an explicit formula:

$a_n=3(-2)^{n-1}$

2) Since a recursive formula, always recur to the previous term let's find the 1st term:

$\begin{gathered} a_n=3(-2)^{n-1} \\ a_1=3(-2)^0 \\ a_1=3 \end{gathered}$

Now the second term:

$\begin{gathered} a_2=3(-2)^{2-1} \\ a_2=-6 \\ \end{gathered}$

Comparing them we can write the Recursive one:

$\begin{gathered} a_n=-2a_{n-1\text{ }} \\ -6=-2(3) \\ -6\text{ = -6 true} \end{gathered}$

3) Then the answers are:

$\begin{gathered} a_1=3 \\ a_n=-2a_{n-1\text{ }} \end{gathered}$