Question

. Find the recursive formula for the geometric
sequence −
1
9
, −
1
3
, −1, −3, −9,

Answer 1

In a geometric sequence, each term is obtained by multiplying the previous term by a constant ratio, known as the common ratio.

To find the common ratio for the given sequence:

1. The first term \( a_1 \) is \( -\frac{1}{9} \).
2. The second term \( a_2 \) is \( -\frac{1}{3} \).
3. To find the common ratio \( r \), divide the second term by the first term:

\[
r = \frac{a_2}{a_1} = \frac{-\frac{1}{3}}{-\frac{1}{9}} = \frac{1/3}{1/9} = \frac{1}{3} \cdot \frac{9}{1} = 3
\]

This calculation shows that the common ratio \( r \) is 3.

We can verify this ratio for the remaining terms:

\[
\frac{a_3}{a_2} = \frac{-1}{-\frac{1}{3}} = 3
\]
\[
\frac{a_4}{a_3} = \frac{-3}{-1} = 3
\]
\[
\frac{a_5}{a_4} = \frac{-9}{-3} = 3
\]

Since all ratios are equal to \( r = 3\), we confirm that the common ratio is indeed 3.

The recursive formula for a geometric sequence can be written as follows:

\[
a_n = r \cdot a_{n-1}
\]

For this specific sequence, the recursive formula is:

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

With the first term defined as:

\[
a_1 = -\frac{1}{9}
\]

Thus, the complete recursive formula for the sequence is:

\[
\begin{cases}
a_1 = -\frac{1}{9} \\
a_n = 3 \cdot a_{n-1} \quad \text{for } n \geq 2
\end{cases}
\]