Question

write the recursive formula for the sequence -2, 8, -16, 32, ..

Answer 1

To find the recursive formula for the sequence \(-2, 8, -16, 32, \ldots\), we first need to identify the pattern in the sequence.

1. **Identify the pattern**:
- The first term is \(-2\).
- The second term is \(8\) (which is \(-2 \times -4\)).
- The third term is \(-16\) (which is \(8 \times -2\)).
- The fourth term is \(32\) (which is \(-16 \times -2\)).

From this, we can observe that each term is being multiplied by \(-2\).

2. **Define the recursive relationship**:
- Let \(a_n\) be the \(n\)-th term of the sequence.
- The recursive formula can be defined starting from the first term:
\[
a_1 = -2
\]
\[
a_n = -2 \cdot a_{n-1} \quad \text{for } n \geq 2.
\]

So, the recursive formula for the sequence is:

\[
\begin{align*}
a_1 & = -2, \\
a_n & = -2 \cdot a_{n-1} \quad \text{for } n \geq 2.
\end{align*}
\]