Question

Write a recursive formula for the following geometric sequence.

{–1, 3, –9, 27, ...}

Answer 1

A geometric sequence is defined by a constant ratio between consecutive terms.

For the given sequence {–1, 3, –9, 27, ...}, we can observe the following:

- The first term \( a_1 = -1 \).
- The second term \( a_2 = 3 \).
- The third term \( a_3 = -9 \).
- The fourth term \( a_4 = 27 \).

To find the common ratio, we divide a term by its previous term:

\[
\text{Common ratio} = \frac{3}{-1} = -3
\]
\[
\text{Common ratio} = \frac{-9}{3} = -3
\]
\[
\text{Common ratio} = \frac{27}{-9} = -3
\]

Thus, the common ratio \( r = -3 \).

We denote the \( n \)-th term of the sequence by \( a_n \). The recursive formula can then be expressed as:

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

So, the recursive formula for the sequence is:
\[
a_1 = -1, \quad a_n = -3 \cdot a_{n-1} \text{ for } n \geq 2.
\]