Question

What is the recursive formula for this geometric sequence?

4, -12, 36, -108…

Answer 1

To find the recursive formula for the given geometric sequence (4, -12, 36, -108...), we first need to identify the common ratio between consecutive terms.

The first term \( a_1 \) is 4.

To find the common ratio \( r \), we can divide the second term by the first term:

\[
r = \frac{a_2}{a_1} = \frac{-12}{4} = -3
\]

To confirm this ratio, we can check the next pairs of terms:
\[
r = \frac{a_3}{a_2} = \frac{36}{-12} = -3
\]

\[
r = \frac{a_4}{a_3} = \frac{-108}{36} = -3
\]

Since the common ratio \( r \) is consistently -3, we can now write the recursive formula for the sequence.

The recursive formula is typically written in the following format:
\[
a_n = r \cdot a_{n-1} \quad \text{for } n \geq 2
\]

Given that the first term \( a_1 = 4 \), the recursive formula for this geometric sequence is:

\[
a_1 = 4
\]
\[
a_n = -3 \cdot a_{n-1} \quad \text{for } n \geq 2
\]

This defines the sequence recursively.