Notes on Sequences, Difference Tables, and Pattern Finding

  • Sequences are ordered lists of numbers; difference tables help identify patterns and predict terms.
  • Definitions:
    • First difference: Δa<em>n=a</em>n+1−anΔa<em>n = a</em>{n+1} - a_n
    • Second difference: Δ2a<em>n=Δa</em>n+1−ΔanΔ^2a<em>n = Δa</em>{n+1} - Δa_n
  • Core Idea: If a difference row becomes constant, the sequence is generated by a polynomial of degree equal to that level (e.g., constant first diffs ⇒\Rightarrow linear; constant second diffs ⇒\Rightarrow quadratic).
    • Consistency in subtraction direction is crucial.
  • Extending the table: Constant differences can be used to reconstruct higher-difference values and predict next terms.
  • Conceptual Takeaway:
    • If Δa<em>nΔa<em>n is constant, a</em>na</em>n is linear: a<em>n=a</em>1+(n−1)da<em>n = a</em>1 + (n-1)d.
    • If Δ2a<em>nΔ^2a<em>n is constant, a</em>na</em>n is quadratic: an=An2+Bn+Ca_n = An^2 + Bn + C.
    • Higher constant differences imply higher-degree polynomial rules.
  • Important Terminology:
    • Explicit (n-term) formula: Directly gives a<em>na<em>n as a function of nn. (e.g., a</em>n=3n−1a</em>n = 3n-1)
    • Recursive (n-term) definition: Defines a<em>na<em>n in terms of previous terms. (e.g., Fibonacci: f</em>n=f<em>n−1+f</em>n−2f</em>n = f<em>{n-1} + f</em>{n-2})
Examples Illustrating the Ideas
  • Example A: Arithmetic-like (2, 5, 8, 11, 14, …)
    • First differences: 3 (constant) ⇒\Rightarrow Linear.
    • Explicit formula: an=2+(n−1)×3=3n−1a_n = 2 + (n-1) \times 3 = 3n - 1.
  • Example B: Quadratic (1, 7, 17, 31, …)
    • First differences: 6, 10, 14, …
    • Second differences: 4, 4, … (constant) ⇒\Rightarrow Quadratic.
    • Explicit formula: an=2n2−1a_n = 2n^2 - 1.
  • Example D: Fibonacci-like (1, 1, 2, 3, 5, 8, …)
    • Recursive definition: f<em>n=f</em>n−1+f<em>n−2f<em>n = f</em>{n-1} + f<em>{n-2} with f</em>1=1,f2=1f</em>1 = 1, f_2 = 1.
    • This sequence does not typically yield constant differences using basic methods.
Recurrence vs Explicit Formulas
  • Recurrence: Good for showing how terms relate; requires prior terms to compute later ones.
  • Explicit: Directly computes any term; better for analysis and prediction.
How to use difference tables to predict the next term
  • Compute differences until a constant row is found.
  • Extend the constant row and propagate values upwards to find the next term in the original sequence.
  • Caveats: Not all sequences have constant differences (e.g., non-polynomial rules like exponential or factorial).