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 45.84 2/5 5/2

1 answer

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:

  1. \( \frac{560}{1400} = \frac{2}{5} \)
  2. \( \frac{224}{560} = \frac{2}{5} \)
  3. \( \frac{89.6}{224} = \frac{2}{5} \)
  4. \( \frac{35.84}{89.6} = \frac{2}{5} \)
  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: \[ a_n = 1400 \cdot \left(\frac{2}{5}\right)^{(n-1)} \]