Question

Write a recursive sequence that represents the sequence defined by the following explicit formula:

an=3−4n

(1 point)
an=an−1−

Answer 1

To define a recursive sequence based on the explicit formula \( a_n = 3 - 4n \), we will express \( a_n \) in terms of \( a_{n-1} \).

First, we can write the explicit formula of \( a_n \) for \( n \) and \( n-1 \):

1. **For \( n \)**:
\[
a_n = 3 - 4n
\]

2. **For \( n-1 \)**:
\[
a_{n-1} = 3 - 4(n - 1) = 3 - 4n + 4 = 7 - 4n
\]

Now we can express \( a_n \) in terms of \( a_{n-1} \):

\[
a_n = a_{n-1} - (a_{n-1} - a_n)
\]

To find \( a_n - a_{n-1} \):
\[
a_n - a_{n-1} = (3 - 4n) - (7 - 4n) = 3 - 4n - 7 + 4n = -4
\]

So we can express the recursive relationship as:

\[
a_n = a_{n-1} - 4
\]

And we need an initial condition. To find \( a_0 \):

\[
a_0 = 3 - 4(0) = 3
\]

Thus, the recursive sequence can be written as:

\[
\begin{align*}
a_n & = a_{n-1} - 4 \quad \text{for } n \geq 1 \\
a_0 & = 3
\end{align*}
\]

In summary, the recursive definition is:

\[
\begin{align*}
a_n & = a_{n-1} - 4 \\
a_0 & = 3
\end{align*}
\]