Let's analyze the sequences and describe their patterns and types:

Sequence I: $-1, 2, 7, 14, \ldots$

• Pattern: Each term increases by an odd number, specifically adding 3, 5, 7, etc.
• Type of Sequence: This is an arithmetic sequence with a second-order difference pattern.

Sequence J: $-5, -\frac{5}{2}, -\frac{5}{4}, -\frac{5}{8}, \ldots$

• Pattern: Each term is multiplied by $\frac{1}{2}$.
• Type of Sequence: This is a geometric sequence with a common ratio of $\frac{1}{2}$.

Sequence K: $6.5, 5, 3.5, 2, \ldots$

• Pattern: Subtract 1.5 each time.
• Type of Sequence: This is an arithmetic sequence with a common difference of $-1.5$.

Sequence L: $86, 85, 83, 80, 76, \ldots$

• Pattern: Decreasing by increasing consecutive integers (-1, -2, -3, etc.).
• Type of Sequence: This is an arithmetic sequence with a non-constant difference pattern.

Sequence M: $-16, 4, -1, -\frac{1}{4}, \ldots$

• Pattern: Dividing by -4 then multiplying by $\frac{1}{4}$.
• Type of Sequence: This is a geometric sequence with alternating positive and negative terms.

Sequence N: $1473.2, 1452.7, 1432.2, 1411.7, \ldots$

• Pattern: Subtracting 20.5 each time.
• Type of Sequence: This is an arithmetic sequence with a common difference of $-20.5$.

Sequence O: $\sqrt{5}, 2, \sqrt{3}, \sqrt{2}, 1, 0, \sqrt{-1}, \ldots$

• Pattern: Decreasing values approaching zero then transitioning to imaginary numbers.
• Type of Sequence: This is a mixed sequence involving radicals and imaginary numbers.

Sequence P: $-4, 12, -36, 108, \ldots$

• Pattern: Alternating multiplication by -3.
• Type of Sequence: This is a geometric sequence with a common ratio of $-3$.

Would you like more details on any specific sequence or have any questions?

Questions to Explore Further:

1. What defines an arithmetic sequence, and how is it different from geometric sequences?
2. How do you identify patterns in sequences with mixed numbers or alternating signs?
3. What are second-order differences, and how do they relate to quadratic sequences?
4. How do imaginary numbers fit into sequences, and what types of patterns involve them?
5. How can you determine the nth term of a geometric sequence?

Tip: For geometric sequences, always look for a multiplicative pattern between terms to identify the common ratio.