To describe the sequence \( \frac{1}{7}, -\frac{5}{21}, \frac{25}{63} \) using the formula \( a_n = a( r )^{n-1} \) where \( a \) is the first term and \( r \) is the common ratio, let's first analyze the terms of the sequence.
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Identify the first term:
- \( a_1 = \frac{1}{7} \)
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Identify the second term:
- \( a_2 = -\frac{5}{21} \)
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Identify the third term:
- \( a_3 = \frac{25}{63} \)
Next, let's check how to find the common ratio \( r \):
- The common ratio can be found by dividing the second term by the first term: \[ r = \frac{a_2}{a_1} = \frac{-\frac{5}{21}}{\frac{1}{7}} = -\frac{5}{21} \times \frac{7}{1} = -\frac{5 \times 7}{21 \times 1} = -\frac{35}{21} = -\frac{5}{3} \]
Now, let's check the ratio between the third and second terms: \[ r = \frac{a_3}{a_2} = \frac{\frac{25}{63}}{-\frac{5}{21}} = \frac{25}{63} \times -\frac{21}{5} = -\frac{25 \times 21}{63 \times 5} = -\frac{525}{315} = -\frac{5}{3} \]
Both calculations give \( r = -\frac{5}{3} \).
Now we can plug in the values into the sequence formula:
- Since \( a = \frac{1}{7} \) and \( r = -\frac{5}{3} \):
So, the equation that describes the sequence is: \[ a_n = \frac{1}{7} \left( -\frac{5}{3} \right)^{n-1} \]
This is the final formula for the given sequence.