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).
3, comma, 1, comma, one third, comma, dot dot dot
3,1,
3
1
​
,...
Find the 7th term.
Find the 7th term.

Answer 1

To analyze the sequence given: \(3, 1, \frac{1}{3}, \ldots\), we need to identify a pattern.

The first three terms are:
- First term: \( a_1 = 3 \)
- Second term: \( a_2 = 1 \)
- Third term: \( a_3 = \frac{1}{3} \)

Next, let's observe the relationship between these terms:

1. From \(3\) to \(1\):
\[
a_2 = \frac{a_1}{3} = \frac{3}{3} = 1
\]

2. From \(1\) to \(\frac{1}{3}\):
\[
a_3 = \frac{a_2}{3} = \frac{1}{3}
\]

We see that each term is derived by dividing the previous term by \(3\). This gives us a recursive formula:
\[
a_n = \frac{a_{n-1}}{3}
\]

Now we can compute the subsequent terms.

- Fourth term:
\[
a_4 = \frac{a_3}{3} = \frac{\frac{1}{3}}{3} = \frac{1}{9}
\]

- Fifth term:
\[
a_5 = \frac{a_4}{3} = \frac{\frac{1}{9}}{3} = \frac{1}{27}
\]

- Sixth term:
\[
a_6 = \frac{a_5}{3} = \frac{\frac{1}{27}}{3} = \frac{1}{81}
\]

- Seventh term:
\[
a_7 = \frac{a_6}{3} = \frac{\frac{1}{81}}{3} = \frac{1}{243}
\]

Now, let’s convert \(\frac{1}{243}\) to a decimal.
Calculating:
\[
\frac{1}{243} \approx 0.00411522633744856...
\]
Rounding to the nearest thousandth, we get:
\[
\approx 0.004
\]

Thus, the 7th term of the sequence is approximately:

\[
\boxed{0.004}
\]