Question

A sequence is explicitly defined as
an = -12 + 4n . If represents the term number, and
is an integer greater than or equal to
, what is the recursive form of the sequence?

Answer 1

To convert the explicit formula of the sequence \( a_n = -12 + 4n \) into a recursive form, we first need to identify the first term and how to relate each term to the previous one.

1. **Finding the first term**:
- For \( n = 1 \):
\[
a_1 = -12 + 4(1) = -12 + 4 = -8
\]

2. **Finding a relationship between successive terms**:
- We can express \( a_n \) in terms of \( a_{n-1} \):
\[
a_n = -12 + 4n
\]
And, for \( n-1 \):
\[
a_{n-1} = -12 + 4(n-1) = -12 + 4n - 4 = -16 + 4n
\]
Now, if we want \( a_n \) in terms of \( a_{n-1} \):
\[
a_n = a_{n-1} + 4
\]

3. **Putting it all together**:
The recursive definition starts with the first term we found and the relationship we derived:
- Initial condition: \( a_1 = -8 \)
- Recursive relation: \( a_n = a_{n-1} + 4 \) for \( n > 1 \)

Therefore, the recursive form of the sequence is:
\[
\begin{align*}
a_1 & = -8 \\
a_n & = a_{n-1} + 4 \quad \text{for } n > 1
\end{align*}
\]