Question

What is the recursive formula for this geometric sequence?
2, -10, 50, -250,..

Answer 1

To find the recursive formula for the given geometric sequence \(2, -10, 50, -250, \ldots\), we first need to identify the common ratio.

To determine the common ratio (\(r\)), we can take any term and divide it by its preceding term:

1. From the first term to the second term:
\[
r = \frac{-10}{2} = -5
\]

2. From the second term to the third term:
\[
r = \frac{50}{-10} = -5
\]

3. From the third term to the fourth term:
\[
r = \frac{-250}{50} = -5
\]

Thus, the common ratio \(r\) is consistently \(-5\).

The first term \(a_1\) of the sequence is \(2\).

A geometric sequence can be defined recursively as:
\[
a_n = r \cdot a_{n-1}
\]
For this sequence, the recursive formula can be defined as follows:

- \(a_1 = 2\)
- \(a_n = -5 \cdot a_{n-1}\) for \(n \geq 2\)

Therefore, the complete recursive formula for the sequence is:
\[
\begin{align*}
a_1 & = 2, \\
a_n & = -5 \cdot a_{n-1} \quad \text{for } n \geq 2.
\end{align*}
\]