To find the explicit formula for the given sequence \( a_n \):
The terms are:
- \( a_1 = 1400 \)
- \( a_2 = 560 \)
- \( a_3 = 224 \)
- \( a_4 = 89.6 \)
- \( a_5 = 35.84 \)
- \( a_6 = 14.336 \)
Let's observe the pattern in the sequence:
Going from \( a_1 \) to \( a_2 \): \[ \frac{560}{1400} = \frac{560 \cdot 10}{1400 \cdot 10} = \frac{56}{140} = \frac{2}{5} \]
Going from \( a_2 \) to \( a_3 \): \[ \frac{224}{560} = 0.4 = \frac{2}{5} \text{ (approx)} \]
So it seems to be multiplying by some factor between each term. Let's check the ratios:
- \( \frac{560}{1400} = \frac{2}{5} \)
- \( \frac{224}{560} = \frac{2}{5} \)
- \( \frac{89.6}{224} = \frac{2}{5} \)
- \( \frac{35.84}{89.6} = \frac{2}{5} \)
- \( \frac{14.336}{35.84} = \frac{2}{5} \)
Thus, we are consistently multiplying each term by \(\frac{2}{5}\).
This means the general form would be: \[ a_n = a_1 \cdot \left(\frac{2}{5}\right)^{(n-1)} \] Substituting \( a_1 = 1400 \): \[ a_n = 1400 \cdot \left(\frac{2}{5}\right)^{(n-1)} \]
Now we can fill in the response areas:
- For the first response area, it will have \( 1400 \).
- For the second response area, it will contain \( \left(\frac{2}{5}\right) \).
So the complete explicit formula for the sequence is: \[ a_n = 1400 \cdot \left(\frac{2}{5}\right)^{(n-1)} \]