Question

Use the initial term and the recursive formula to find an explicit formula for the sequence an. Write your answer in simplest form.
a1
=
–
4
an
=
–
an–1
an=

Answer 1

To find an explicit formula for the sequence defined by the initial term \( a_1 = -4 \) and the recursive formula \( a_n = -a_{n-1} \), we can compute a few terms of the sequence and look for a pattern.

1. Given \( a_1 = -4 \)
2. Using the recursive formula for \( n=2 \):
\[
a_2 = -a_1 = -(-4) = 4
\]
3. For \( n=3 \):
\[
a_3 = -a_2 = -4
\]
4. For \( n=4 \):
\[
a_4 = -a_3 = -(-4) = 4
\]
5. For \( n=5 \):
\[
a_5 = -a_4 = -4
\]
6. For \( n=6 \):
\[
a_6 = -a_5 = -(-4) = 4
\]

Now we notice a pattern from our calculations:
- For odd \( n \) (1, 3, 5, ...), \( a_n = -4 \)
- For even \( n \) (2, 4, 6, ...), \( a_n = 4 \)

Based on this pattern, we can derive an explicit formula. We can express this using \( n \) as follows:

\[
a_n =
\begin{cases}
-4 & \text{if } n \text{ is odd} \\
4 & \text{if } n \text{ is even}
\end{cases}
\]

This can be expressed in a more compact form using the expression:
\[
a_n = (-1)^n \cdot 4
\]

This matches our earlier findings:
- When \( n \) is odd, \( (-1)^n = -1 \), so \( a_n = -4 \).
- When \( n \) is even, \( (-1)^n = 1 \), so \( a_n = 4 \).

Thus, the explicit formula for the sequence \( a_n \) is:

\[
\boxed{(-1)^n \cdot 4}
\]