The sequence in the image shows the following numbers: $9, 3, 1, \frac{1}{3}, \dots$. We are tasked with finding the next term in the sequence as well as $S_6$, which seems to indicate the sum of the first six terms.

Step 1: Recognizing the Pattern

The given sequence appears to be geometric because the ratio between consecutive terms is constant.

Let's verify:

• From $9$ to $3$: $\frac{3}{9} = \frac{1}{3}$
• From $3$ to $1$: $\frac{1}{3} = \frac{1}{3}$
• From $1$ to $\frac{1}{3}$: $\frac{\frac{1}{3}}{1} = \frac{1}{3}$

Thus, the common ratio $r = \frac{1}{3}$.

Step 2: Finding the Next Term

To find the next term $a_5$:

$a_5 = \frac{1}{3} \times \frac{1}{3} = \frac{1}{9}$

Step 3: Finding $S_6$

The sum $S_6$ of the first six terms in a geometric series can be calculated using the formula for the sum of the first $n$ terms:

$S_n = a_1 \frac{1 - r^n}{1 - r}$

Here, $a_1 = 9$, $r = \frac{1}{3}$, and $n = 6$:

$S_6 = 9 \times \frac{1 - \left(\frac{1}{3}\right)^6}{1 - \frac{1}{3}}$

Let's calculate $S_6$.The sum of the first six terms of the geometric series $S_6$ is approximately $13.48$.

Summary:

1. The next term in the sequence $a_5$ is $\frac{1}{9}$.
2. The sum of the first six terms $S_6$ is approximately $13.48$.

Do you want further clarification or more details?

Here are five related questions to expand on this topic:

1. How do you calculate the sum of an infinite geometric series?
2. What conditions make a geometric series converge?
3. How would the sum $S_n$ change if the common ratio $r$ were negative?
4. How can you apply the geometric series formula in financial problems?
5. What are the applications of geometric sequences in nature?

Tip: In geometric sequences, the ratio between terms is always constant. This makes them easy to identify and work with when solving problems.