Question

8. A sequence is shown.

1400, 560, 224, 89.6, 35,84,.....

Complete the explicit formula for the sequence.

(2 points)
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
an=
Response area. Response area(n−1)
1400,35.84,2/5,5/2

Answer 1

To find the explicit formula for the given sequence \(1400, 560, 224, 89.6, 35.84, \ldots\), we can first look for a pattern.

Let's analyze the ratios between consecutive terms:

- \( \frac{560}{1400} = \frac{2}{5} \)
- \( \frac{224}{560} = \frac{2}{5} \)
- \( \frac{89.6}{224} = \frac{2}{5} \)
- \( \frac{35.84}{89.6} = \frac{2}{5} \)

From above, we can see that each term of the sequence is obtained by multiplying the previous term by \( \frac{2}{5} \).

Starting with \( a_1 = 1400 \):

- \( a_2 = a_1 \cdot \frac{2}{5} \)
- \( a_3 = a_2 \cdot \frac{2}{5} = a_1 \cdot \left(\frac{2}{5}\right)^2 \)
- \( a_4 = a_3 \cdot \frac{2}{5} = a_1 \cdot \left(\frac{2}{5}\right)^3 \)
- Continuing this pattern, we have \( a_n = 1400 \cdot \left(\frac{2}{5}\right)^{n-1} \)

Thus, the explicit formula for the sequence is:

\[
a_n = 1400 \cdot \left(\frac{2}{5}\right)^{n-1}
\]

So the response for the formula to complete is:

**an = 1400 \cdot (2/5)^(n-1)**