Question

1400, 560, 224, 89.6, 35, 84,.....
Complete the explicit formula for the sequence. (4 points)
an= _______ . _______(n−1)

Answer 1

To find the explicit formula for the given sequence \(1400, 560, 224, 89.6, 35, 84, \ldots\), we can start by examining how the terms relate to each other.

Let's look at the ratios of consecutive terms:

\[
\frac{560}{1400} = 0.4
\]
\[
\frac{224}{560} = 0.4
\]
\[
\frac{89.6}{224} = 0.4
\]
\[
\frac{35}{89.6} = 0.4
\]
\[
\frac{84}{35} = 2.4
\]

The first four terms appear to be multiplied by a common ratio of 0.4, but then we see that the last term does not follow this ratio.

Given the pattern, it seems that we have two different behaviors within the sequence. The initial terms (up to 35) decrease by a factor of 0.4, while the last term (84) appears to be out of sync.

However, if we take the first five terms, \(1400, 560, 224, 89.6, 35\), we can denote them by an explicit formula.

If we define \(a_n\) for \(n = 1, 2, 3, 4, 5\):
- \(a_1 = 1400\)
- \(a_2 = 560\)
- \(a_3 = 224\)
- \(a_4 = 89.6\)
- \(a_5 = 35\)

The common ratio between the first four terms is \(0.4\).

The formula then could be expressed as:
\[
a_n = 1400 \cdot (0.4)^{n-1} \quad \text{for } n = 1, 2, 3, 4, 5
\]

To complete your equation in the required form:
\[
a_n = 1400 \cdot (0.4)^{n-1}
\]

So the answer to your explicit formula is:
\[
a_n = 1400 \cdot 0.4^{n - 1}
\]