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).
5, comma, 2, comma, start fraction, 4, divided by, 5, end fraction, comma, dot dot dot
5,2,
5
4
,...
Find the 10th term.
Find the 10th term
5, comma, 2, comma, start fraction, 4, divided by, 5, end fraction, comma, dot dot dot
5,2,
5
4
,...
Find the 10th term.
Find the 10th term
Answers
GPT-4o mini
To find the 10th term of the sequence given by the first three terms \(5, 2, \frac{4}{5}, \ldots\), we first need to see if there is a pattern.
1. **Identify the pattern**:
- The first term is \(5\).
- The second term is \(2\).
- The third term is \(\frac{4}{5}\).
One possible approach to obtaining the next terms is to analyze how the terms relate to each other:
- From \(5\) to \(2\), the difference is \(5 - 2 = 3\).
- From \(2\) to \( \frac{4}{5}\), we can express this in decimal form: \(2 = \frac{10}{5}\) leads to the difference of \(2 - \frac{4}{5} = \frac{10}{5} - \frac{4}{5} = \frac{6}{5} = 1.2\).
As there are no obvious arithmetic or geometric relations, let's analyze the pattern in terms of fractions:
- It’s observed that \(5 = \frac{25}{5}\), \(2 = \frac{10}{5}\), and \( \frac{4}{5}\).
The first three terms appear to decrease in a pattern of divisions by \(5\) after the first term, but let’s find if we can represent this in a formula.
2. **Finding a general formula**:
To find subsequent terms, let’s denote the \(n\)-th term as \(a_n\):
- Initial observations:
- \(a_1 = 5\)
- \(a_2 = 2\)
- \(a_3 = \frac{4}{5} = 0.8\)
Let's observe the decreasing nature mixed with division. If this sequence can be seen iteratively halving:
- \(5\) to \(2\) seems halving and scaling \(4\) yields \(2\).
- Then \(2\) to \(0.8\) could involve something like \(\frac{4}{5} = 2 \times \frac{2}{5}\).
The term can be defined recursively, but we also can check explicitly as:
\[
a_4 = \frac{4}{5} \cdot \frac{1}{2}=(0.8)(0.5)=0.4,
\]
\[
a_5 = 0.4(0.5)=0.2,
\]
\[
a_6 = 0.2(0.5)=0.1,
\]
\[
a_7 = 0.1(0.5)=0.05,
\]
Now it appears to halve after \(5\).
Through checking, this does seem exponential from term \(4\).
To find the \(10\)-th, following from \(a_4,\) recursively continuing:
\[
a_8 = \frac{0.05}{2} = 0.025,
\]
\[
a_9 = 0.025/2 \approx 0.0125,
\]
\[
a_{10} = 0.0125/2 \approx 0.00625.
\]
3. **Final calculation**:
To round this value to the nearest thousandth:
- The \(10^{th}\) term \(a_{10} = 0.00625\) rounds to \(0.006\) when rounding to \(3\) decimal places.
Thus, the result is:
\[
\boxed{0.006}
\]
That would be the \(10^{th}\) term, rounded off correctly.
1. **Identify the pattern**:
- The first term is \(5\).
- The second term is \(2\).
- The third term is \(\frac{4}{5}\).
One possible approach to obtaining the next terms is to analyze how the terms relate to each other:
- From \(5\) to \(2\), the difference is \(5 - 2 = 3\).
- From \(2\) to \( \frac{4}{5}\), we can express this in decimal form: \(2 = \frac{10}{5}\) leads to the difference of \(2 - \frac{4}{5} = \frac{10}{5} - \frac{4}{5} = \frac{6}{5} = 1.2\).
As there are no obvious arithmetic or geometric relations, let's analyze the pattern in terms of fractions:
- It’s observed that \(5 = \frac{25}{5}\), \(2 = \frac{10}{5}\), and \( \frac{4}{5}\).
The first three terms appear to decrease in a pattern of divisions by \(5\) after the first term, but let’s find if we can represent this in a formula.
2. **Finding a general formula**:
To find subsequent terms, let’s denote the \(n\)-th term as \(a_n\):
- Initial observations:
- \(a_1 = 5\)
- \(a_2 = 2\)
- \(a_3 = \frac{4}{5} = 0.8\)
Let's observe the decreasing nature mixed with division. If this sequence can be seen iteratively halving:
- \(5\) to \(2\) seems halving and scaling \(4\) yields \(2\).
- Then \(2\) to \(0.8\) could involve something like \(\frac{4}{5} = 2 \times \frac{2}{5}\).
The term can be defined recursively, but we also can check explicitly as:
\[
a_4 = \frac{4}{5} \cdot \frac{1}{2}=(0.8)(0.5)=0.4,
\]
\[
a_5 = 0.4(0.5)=0.2,
\]
\[
a_6 = 0.2(0.5)=0.1,
\]
\[
a_7 = 0.1(0.5)=0.05,
\]
Now it appears to halve after \(5\).
Through checking, this does seem exponential from term \(4\).
To find the \(10\)-th, following from \(a_4,\) recursively continuing:
\[
a_8 = \frac{0.05}{2} = 0.025,
\]
\[
a_9 = 0.025/2 \approx 0.0125,
\]
\[
a_{10} = 0.0125/2 \approx 0.00625.
\]
3. **Final calculation**:
To round this value to the nearest thousandth:
- The \(10^{th}\) term \(a_{10} = 0.00625\) rounds to \(0.006\) when rounding to \(3\) decimal places.
Thus, the result is:
\[
\boxed{0.006}
\]
That would be the \(10^{th}\) term, rounded off correctly.