https://file.aiquickdraw.com/mathbot/file/1726468992519we22qo7m.jpg

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.

What is the next term in the geometric sequence 9, 3, 1, 1/3, ...? Also, calculate the sum of the first six terms (S6).

Learn how to solve a geometric sequence problem. Find the next term in the sequence 9, 3, 1, 1/3, ... and calculate the sum of the first six terms (S6). Detailed solution using the geometric series sum formula. Suitable for students in Grades 9-11.

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