Question

A sequence is shown
1400,560,224,89.6,35,84…..
Complete the explicit formula for the sequence
_________*____(n-1)
Options
1400, 35.84, 2/5, 5/2

Answer 1

To find the explicit formula for the given sequence \(1400, 560, 224, 89.6, 35, 84,\) we first need to observe the pattern of the terms:

1. **Identify the sequence:**
- \(1400\)
- \(560 = 1400 \times 0.4\)
- \(224 = 560 \times 0.4\)
- \(89.6 = 224 \times 0.4\)
- \(35 = 89.6 \times 0.4\)
- $84 = 35 \times 2.4$

From our observations, we can see that the first five terms seem to follow a multiplication by \(0.4\), and then it jumps to multiplying by \(2.4\) for the last term, indicating a potential pattern change.

2. **Finding a formula:**
The general pattern for the first five terms appears to be:

\[
a_n = 1400 \times 0.4^{(n-1)} \quad \text{for } n \in \{1, 2, 3, 4, 5\}
\]

However, we need to address the last term as well. We can represent the entire sequence as follows:

For \(n = 1\) to \(5\):
\[
a_n = 1400 \cdot (0.4)^{(n-1)}
\]

For \(n = 6\):
The sequence can be adjusted, or we can recognize the pattern that \(a_6 = 35 \times 2.4\), meaning we might only use part of this formula for initial terms and adjust accordingly for \(n = 6\).

3. **Explicit formula:**
Considering the question's request for an explicit formula, focusing primarily on the geometric part of the sequence, we can say:

\[
a_n = 1400 \cdot (0.4)^{(n-1)} \text{ for } n = 1, 2, 3, 4, 5
\]
\[
a_6 = 35 \cdot 2.4
\]

Thus, filling in the blanks for the explicit formula given in the form \( \text{_________} * \text{____}(n-1) \):

The best straightforward structure from what we've observed is:

**Final formula:**
\[
a_n = 1400 * (0.4)^{(n-1)}
\]

So the answer is:
- The first blank is **1400**
- The second blank is **0.4**

Hence, the final formula is:
\[
1400 * (0.4)^{(n-1)}
\]

Please let me know if you need anything else!