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).
24, comma, 16, comma, start fraction, 32, divided by, 3, end fraction, comma, point, point, .
24,16,
3
32
​
,...
Find the 9th term.
Find the 9th term.

Answer 1

To find the 9th term of the sequence given by the first three terms: 24, 16, and \( \frac{32}{3} \), we need to observe the pattern or rule governing the sequence.

1. **Identify the terms**:
- 1st term: \( a_1 = 24 \)
- 2nd term: \( a_2 = 16 \)
- 3rd term: \( a_3 = \frac{32}{3} \approx 10.667 \)

2. **Check the ratios between terms**:
- \( \frac{a_2}{a_1} = \frac{16}{24} = \frac{2}{3} \)
- \( \frac{a_3}{a_2} = \frac{\frac{32}{3}}{16} = \frac{32}{48} = \frac{2}{3} \)

This suggests a pattern based on multiplication by \( \frac{2}{3} \).

3. **Find the formula for the \( n \)th term**:
The sequence appears to be a geometric sequence where:
\[
a_n = a_1 \left( \frac{2}{3} \right)^{n-1}
\]

4. **Substituting for the 9th term**:
\[
a_9 = 24 \left( \frac{2}{3} \right)^{8}
\]

5. **Calculating \( a_9 \)**:
First, calculate \( \left( \frac{2}{3} \right)^{8} \):
\[
\left( \frac{2}{3} \right)^8 = \frac{2^8}{3^8} = \frac{256}{6561}
\]

Now multiply by 24:
\[
a_9 = 24 \times \frac{256}{6561} = \frac{6144}{6561}
\]

Calculate this as a decimal:
\[
\frac{6144}{6561} \approx 0.936
\]

In conclusion, the 9th term of the sequence rounded to the nearest thousandth is:

\[
\boxed{0.936}
\]