What is the formula that describes the sequence 8, 5, 2, 1, 4?

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The given sequence is 8, 5, 2, 1, 4. To find a formula that describes this sequence, we first need to analyze the relationship between the terms.

1. **Identifying the Sequence Pattern:**

  • We can observe that the sequence does not follow a simple arithmetic or geometric pattern.
  • By examining the differences between consecutive terms, we can see that the differences are: 8 – 5 = 3, 5 – 2 = 3, 2 – 1 = 1, and 1 – 4 = -3.

2. **Defining the Terms in the Sequence:**

  • Let’s denote the terms of this sequence as an where n represents the position of the term in the sequence (starting from 1).
  • Thus, we have:
    • a1 = 8
    • a2 = 5
    • a3 = 2
    • a4 = 1
    • a5 = 4

3. **Finding a Formula:**

Looking closely at the terms, it appears the sequence might represent alternating positive and negative changes:

  • The sequence decreases from 8 to 5 then to 2, but it makes a drop to 1 before rising to 4.
  • The terms do not fit neatly into a standard polynomial or series form, but we can define a piecewise function or recursive relation instead.

4. **Proposing a Recursive Formula:**

A possible recursive formula for this sequence could be:

a1 = 8
a2 = a1 - 3
a3 = a2 - 3
a4 = a3 - 1
a5 = a4 + 3

In this case, the recursive approach allows us to define subsequent terms based on a defined set of operations. However, since the sequence does not have a constant change or straightforward fit in polynomial expressions, we finalize with:

5. **Conclusion:**

While an explicit formula in the traditional sense might not be readily apparent, using the identified pattern and recursive relations enables us to represent the sequence reasonably. To summarize:

an = starting	erm + adjustments based on previous terms.

Thus, using the recursive relation is one effective way to describe and work with the sequence 8, 5, 2, 1, 4.

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