To find the explicit formula for the given sequence \(1400, 560, 224, 89.6, 35, 84,\) we first need to observe the pattern of the terms:
- Identify the sequence:
- \(1400\)
- \(560 = 1400 \times 0.4\)
- \(224 = 560 \times 0.4\)
- \(89.6 = 224 \times 0.4\)
- \(35 = 89.6 \times 0.4\)
- $84 = 35 \times 2.4$
From our observations, we can see that the first five terms seem to follow a multiplication by \(0.4\), and then it jumps to multiplying by \(2.4\) for the last term, indicating a potential pattern change.
- Finding a formula: The general pattern for the first five terms appears to be:
\[ a_n = 1400 \times 0.4^{(n-1)} \quad \text{for } n \in {1, 2, 3, 4, 5} \]
However, we need to address the last term as well. We can represent the entire sequence as follows:
For \(n = 1\) to \(5\): \[ a_n = 1400 \cdot (0.4)^{(n-1)} \]
For \(n = 6\): The sequence can be adjusted, or we can recognize the pattern that \(a_6 = 35 \times 2.4\), meaning we might only use part of this formula for initial terms and adjust accordingly for \(n = 6\).
- Explicit formula: Considering the question's request for an explicit formula, focusing primarily on the geometric part of the sequence, we can say:
\[ a_n = 1400 \cdot (0.4)^{(n-1)} \text{ for } n = 1, 2, 3, 4, 5 \] \[ a_6 = 35 \cdot 2.4 \]
Thus, filling in the blanks for the explicit formula given in the form \( \text{_____} * \text{}(n-1) \):
The best straightforward structure from what we've observed is:
Final formula: \[ a_n = 1400 * (0.4)^{(n-1)} \]
So the answer is:
- The first blank is 1400
- The second blank is 0.4
Hence, the final formula is: \[ 1400 * (0.4)^{(n-1)} \]
Please let me know if you need anything else!
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