To derive a formula for the sequence -69, 138, -276, we first need to identify a pattern in the sequence.
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Identify the first term:
- The first term (when n = 1) is \( a_1 = -69 \).
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Identify the pattern between consecutive terms:
- The second term (n = 2) is \( a_2 = 138 \).
- The third term (n = 3) is \( a_3 = -276 \).
- We can see that the sequence alternates in sign and appears to increase in absolute value.
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Calculate the ratios of consecutive terms:
- \( \frac{a_2}{a_1} = \frac{138}{-69} = -2 \)
- \( \frac{a_3}{a_2} = \frac{-276}{138} = -2 \)
This indicates that each term is -2 times the previous term.
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Identify the general form of the sequence:
- The sequence can be defined using an exponential function based on the previous calculations.
The general term can be written as: \[ a_n = a_1 \cdot r^{(n-1)} \] where \( r \) is the common ratio.
Substituting the known values:
- \( a_1 = -69 \)
- \( r = -2 \)
Therefore, the equation that describes the sequence is: \[ a_n = -69 \cdot (-2)^{(n-1)} \]
So, you can fill in the answer as: \[ a_n = -69 \cdot (-2)^{n-1} \]