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).
6, comma, start fraction, 12, divided by, 5, end fraction, comma, start fraction, 24, divided by, 25, end fraction, comma, dot dot dot
6,
5
12
​
,
25
24
​
,...
Find the 7th term.
Find the 7th term.

Answer 1

To find the 7th term of the sequence given the first three terms \(6\), \(\frac{12}{5}\), and \(\frac{24}{25}\), we can first look for a pattern in the sequence.

1. The first term is \(6\).
2. The second term is \(\frac{12}{5}\).
3. The third term is \(\frac{24}{25}\).

Next, we can also express these terms in decimal form:
- \(6 = 6.000\)
- \(\frac{12}{5} = 2.4\)
- \(\frac{24}{25} = 0.96\)

Now let's observe the relationship between the terms. Notice the following:

- From \(6\) to \(\frac{12}{5}\):
\[
\frac{12}{5} = 6 \cdot \frac{2}{5}
\]

- From \(\frac{12}{5}\) to \(\frac{24}{25}\):
\[
\frac{24}{25} = \frac{12}{5} \cdot \frac{2}{5}
\]

It can be inferred that the ratio of each term to the previous term is \(\frac{2}{5}\). Let's confirm that by looking at the ratios:
1. \(\frac{\frac{12}{5}}{6} = \frac{12/5}{30/5} = \frac{12}{30} = \frac{2}{5}\)
2. \(\frac{\frac{24}{25}}{\frac{12}{5}} = \frac{24/25}{12/5} = \frac{24 \cdot 5}{12 \cdot 25} = \frac{120}{300} = \frac{2}{5}\)

Now, we can find the next terms using the ratio:
- 4th term:
\[
\frac{24}{25} \cdot \frac{2}{5} = \frac{48}{125}
\]

- 5th term:
\[
\frac{48}{125} \cdot \frac{2}{5} = \frac{96}{625}
\]

- 6th term:
\[
\frac{96}{625} \cdot \frac{2}{5} = \frac{192}{3125}
\]

- 7th term:
\[
\frac{192}{3125} \cdot \frac{2}{5} = \frac{384}{15625}
\]

Now, let's convert the 7th term \(\frac{384}{15625}\) into decimal and round to the nearest thousandth:
\[
\frac{384}{15625} \approx 0.024576
\]

Rounding to the nearest thousandth, we get \(0.025\).

Thus, the 7th term is:
\[
\boxed{0.025}
\]