This sequence appears to follow a non-linear pattern. To identify a potential rule, let’s analyze the differences between consecutive terms.

Sequence:

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

Step 1: Differences between consecutive terms

1. $0 - 7 = -7$
2. $-1 - 0 = -1$
3. $-1 - (-1) = 0$
4. $-2 - (-1) = -1$
5. $-1 - (-2) = 1$

Step 2: Identify pattern

The differences are varying, suggesting the sequence doesn't follow a simple arithmetic progression. We could try exploring other kinds of patterns, such as geometric progression or recursive relationships, but additional terms or a rule is needed to define a clear formula.

Hypothesis:

The transition from integer values to fractional values suggests a potential change in the rule at some point in the sequence.

Would you like to explore this sequence further by testing possible patterns or providing more terms? Additionally, would you like me to dive deeper into specific methods to analyze this?

Follow-up Questions:

1. Do you think this sequence involves a recursive rule?
2. Is there a step where fractions or ratios begin to dominate the pattern?
3. Could the earlier values represent part of a piecewise function?
4. Should I check for polynomial relationships to predict further terms?
5. Could this be a special sequence related to a particular problem or theory?

Tip:

Analyzing the first or second differences between terms can often reveal hidden patterns, especially in non-linear sequences.