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 express the given explicit formula \( a_n = 3 - 4n \) as a recursive sequence, we need to establish a relationship between consecutive terms.

First, let's compute the first few values based on the explicit formula:

- \( a_0 = 3 - 4(0) = 3 \)
- \( a_1 = 3 - 4(1) = 3 - 4 = -1 \)
- \( a_2 = 3 - 4(2) = 3 - 8 = -5 \)
- \( a_3 = 3 - 4(3) = 3 - 12 = -9 \)

From these values, we can notice a pattern. The difference between consecutive terms can help us formulate the recursive relationship:

- \( a_1 - a_0 = -1 - 3 = -4 \)
- \( a_2 - a_1 = -5 - (-1) = -4 \)
- \( a_3 - a_2 = -9 - (-5) = -4 \)

The difference \( a_n - a_{n-1} \) is constant and equals \(-4\). Thus, we can express the recursive sequence as follows:

- Base case: \( a_0 = 3 \)
- Recursive case: \( a_n = a_{n-1} - 4 \) for \( n \geq 1 \)

Putting this together, the recursive sequence can be written as:

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