Question

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, dot dot dot
36,27,
4
81
​
,...
Find the 10th term.

Answer 1

To find the 10th term of the sequence given the first three terms: \( a_1 = 36 \), \( a_2 = 27 \), and \( a_3 = \frac{81}{4} \).

### Step 1: Finding a pattern

First, we will calculate the ratios between consecutive terms and see if there’s a common ratio (indicating that this might be a geometric sequence).

1. **Calculate** \( r_1 = \frac{a_2}{a_1} = \frac{27}{36} = \frac{3}{4} \)
2. **Calculate** \( r_2 = \frac{a_3}{a_2} = \frac{\frac{81}{4}}{27} = \frac{81}{4 \times 27} = \frac{81}{108} = \frac{3}{4} \)

Since both ratios are \( \frac{3}{4} \), we can infer that this is a geometric sequence with a common ratio of \( r = \frac{3}{4} \).

### Step 2: Finding the 10th term

The general formula for the n-th term of a geometric sequence is given by:

\[
a_n = a_1 \cdot r^{(n-1)}
\]

Where:
- \( a_1 = 36 \)
- \( r = \frac{3}{4} \)
- \( n = 10 \)

Substituting these values into the formula:

\[
a_{10} = 36 \cdot \left(\frac{3}{4}\right)^{9}
\]

### Step 3: Calculating \( a_{10} \)

Now we need to compute \( \left(\frac{3}{4}\right)^{9} \):

\[
\left(\frac{3}{4}\right)^{9} = \frac{3^9}{4^9} = \frac{19683}{262144}
\]

Now, we calculate \( a_{10} \):

\[
a_{10} = 36 \cdot \frac{19683}{262144}
\]

Calculating \( 36 \cdot 19683 \):

\[
36 \cdot 19683 = 710568
\]

Now, we divide this by \( 262144 \):

\[
a_{10} = \frac{710568}{262144} \approx 2.709
\]

### Final Result

Rounding to the nearest thousandth:

\[
\boxed{2.709}
\]