The first three terms of a sequence are given. Write your answer as a decimal or whole number. Round to the nearest thousandth (if necessary).
36, comma, 27, comma, start fraction, 81, divided by, 4, end fraction, comma, point, point, .
36,27, 
4
81
​
 ,...
Find the 10th term.
Find the 10th term.

To find the 10th term of the sequence \(36, 27, \frac{81}{4}, \dots\), we first need to analyze the pattern.

### Step 1: Identify the pattern

Let's check the ratio between consecutive terms to see if this is a geometric sequence:

- The ratio of the 2nd term to the 1st term: 
  \[
  \frac{27}{36} = \frac{3}{4}
  \]
  
- The ratio of the 3rd term to the 2nd term: 
  \[
  \frac{\frac{81}{4}}{27} = \frac{81}{4} \times \frac{1}{27} = \frac{81}{108} = \frac{3}{4}
  \]

Since the ratio between consecutive terms is consistent (\(\frac{3}{4}\)), this confirms that the sequence is geometric with a common ratio \(r = \frac{3}{4}\).

### Step 2: General formula for the nth term of a geometric sequence

The formula for the nth term of a geometric sequence is given by:
\[
a_n = a_1 \times r^{n-1}
\]
where:
- \(a_1\) is the first term,
- \(r\) is the common ratio, and
- \(n\) is the term number.

### Step 3: Calculate the 10th term

Using the given values:
\[
a_1 = 36, \quad r = \frac{3}{4}, \quad n = 10
\]

The 10th term is:
\[
a_{10} = 36 \times \left(\frac{3}{4}\right)^{10-1} = 36 \times \left(\frac{3}{4}\right)^9
\]

Now, let's calculate \(\left(\frac{3}{4}\right)^9\):

\[
\left(\frac{3}{4}\right)^9 \approx 0.075272
\]

Thus, the 10th term is:
\[
a_{10} = 36 \times 0.075272 \approx 2.710
\]

### Final Answer:
The 10th term of the sequence is approximately **2.710**.

Would you like to explore any further details or have any other questions? Here are some related questions:

1. What is the 11th term of this sequence?
2. What is the sum of the first 10 terms of this sequence?
3. How does the sequence behave as the term number increases (does it approach a specific value)?
4. What would the 10th term be if the first term were 48 instead of 36?
5. What is the ratio between the 10th and 9th terms?

**Tip:** When dealing with geometric sequences, calculating a higher power of the common ratio can often be simplified using logarithms or approximation methods to avoid lengthy calculations.

Learn how to find the 10th term of the geometric sequence 36, 27, 81/4, ... using the formula for the nth term of a geometric sequence. Step-by-step solution provided with calculations rounded to the nearest thousandth. Suitable for high school students studying sequences and series.

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