To find the explicit formula for the sequence defined by the initial term \( a_1 = -40 \) and the recursive formula \( a_n = 5a_{n-1} \), we start by calculating the first few terms of the sequence:
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Calculate the first few terms:
- \( a_1 = -40 \)
- \( a_2 = 5a_1 = 5(-40) = -200 \)
- \( a_3 = 5a_2 = 5(-200) = -1000 \)
- \( a_4 = 5a_3 = 5(-1000) = -5000 \)
- \( a_5 = 5a_4 = 5(-5000) = -25000 \)
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From the pattern: We can observe that each term can be expressed as a power of 5 multiplied by the initial term \( a_1 \):
- \( a_2 = 5^1 \cdot a_1 \)
- \( a_3 = 5^2 \cdot a_1 \)
- \( a_4 = 5^3 \cdot a_1 \)
- \( a_5 = 5^4 \cdot a_1 \)
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Generalizing this pattern: It appears that for any \( n \), we have: \[ a_n = 5^{n-1} \cdot a_1 \]
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Substituting the initial value: Since \( a_1 = -40 \), we substitute this into our formula: \[ a_n = 5^{n-1} \cdot (-40) = -40 \cdot 5^{n-1} \]
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Final explicit formula: Thus, the explicit formula for the sequence is: \[ a_n = -40 \cdot 5^{n-1} \]
This formula allows you to calculate any term \( a_n \) in the sequence directly without the need for recursion.
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