To find an explicit formula for the given sequence \(1400, 560, 224, 89.6, 35.84, \ldots\), let's first look at how the terms are changing:
- Finding the pattern:
- The first term is \(1400\).
- The second term is \(1400 \times \frac{2}{5} = 560\).
- The third term is \(560 \times \frac{2}{5} = 224\).
- The fourth term is \(224 \times \frac{2}{5} = 89.6\).
- The fifth term is \(89.6 \times \frac{2}{5} = 35.84\).
From this, we can see that each term can be obtained by multiplying the previous term by \(\frac{2}{5}\).
- Writing the explicit formula:
- The \(n\)-th term can be expressed as: \[ a_n = a_1 \left(\frac{2}{5}\right)^{n-1} \]
- Here, \(a_1 = 1400\).
Thus, the explicit formula for this sequence is: \[ a_n = 1400 \left(\frac{2}{5}\right)^{n-1} \]
This formula defines each term in the sequence based on the position \(n\).
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