This sequence seems to follow a specific pattern or rule, but it isn't immediately clear. Let's break it down step by step to see if we can identify a consistent pattern or relationship between the terms.

Given Sequence:

$7, 0, -1, -1, -2, -1, \dots, -\frac{2}{3}, -\frac{5}{3}, -\frac{8}{9}$

We'll check for differences between consecutive terms first to see if there is an arithmetic or geometric progression.

Differences between terms:

1. $0 - 7 = -7$
2. $-1 - 0 = -1$
3. $-1 - (-1) = 0$
4. $-2 - (-1) = -1$
5. $-1 - (-2) = 1$
6. $-\frac{2}{3} - (-1) = \frac{1}{3}$
7. $-\frac{5}{3} - (-\frac{2}{3}) = -1$
8. $-\frac{8}{9} - (-\frac{5}{3}) = \frac{7}{9}$

There seems to be no simple arithmetic progression between the differences. Additionally, the introduction of fractions suggests a change in the pattern after the integer terms.

Possible Pattern:

This sequence could be a combination of different rules for different sections of the sequence, or a more complex function generating the terms.

Investigating Patterns:

• There could be alternating behaviors for even and odd terms.
• The fractions at the end could represent a switch to a different series.

Next Steps:

Would you like me to attempt to fit a more advanced pattern or use a particular method to identify the rule governing this sequence?

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Relative Questions:

1. Could this sequence involve multiple formulas for different sections?
2. Do you suspect the pattern switches between fractions and integers?
3. Could a recursive formula be involved in this sequence?
4. Would you like to explore geometric progressions as a potential explanation?
5. Should we attempt a polynomial fit for the sequence?

Tip: Sequences can often switch patterns at certain points, so always check for alternating or piecewise behaviors!