Chapter 8: Sequences and Progressions
Explore the mathematics of prediction, linear arithmetic patterns, and exponential geometric behavior.
๐ Core Vocabulary & Definitions
- Sequence: Ordered list of terms (t₁, t₂, t₃, …) following a rule.
- Explicit Rule: Formula calculating any term directly using position (n).
- Recursive Rule: Defines a term using preceding terms (e.g., tn = tn-1 + d).
- Arithmetic Progression (AP): Sequence with constant difference (d).
- Geometric Progression (GP): Sequence with constant ratio (r).
๐ Essential Formulas & Properties
General Term of AP: tn = a + (n - 1)d
Sum of First n Naturals: Sn = n(n + 1)⁄2
General Term of GP: tn = a · rn-1
AP Linear Graph: Plotting (n, tn) yields a perfect straight line.
⚙️ Algorithm: Finding Sequence Rules
Calculate Diff
Evaluate consecutive differences (e.g., 4 - 1 = 3). Constant diff confirms AP.
Recursive Rule
State t₁ and define next: t₁ = 1, tn = tn-1 + 3 (for n ≥ 2).
Explicit Formula
Substitute into tn = a + (n-1)d. Use a=1, d=3 to build formula.
Verify & Simplify
Simplify to tn = 3n - 2. Check with n=1, 2 to ensure accuracy.
⭐ Key Properties & Historical Context
- Indian Metric History: 10-meter uniform street spacing in Sindhu-Sarasvati grid systems shows ancient AP roots.
- Visual Sum Proof: Fitting two identical triangle dot arrays into an n × (n+1) rectangle proves Sn.
- Sierpiลski Fractal: Removing center triangles creates a GP of black triangles with shrinking area.
๐ Quick Reference: AP vs GP
| Feature | AP | GP |
|---|---|---|
| Base Rule | Add constant d | Multiply constant r |
| nth Term | tn = a + (n-1)d | tn = a · rn-1 |
| Graph Shape | Straight Line | Exponential Curve |
⚠️ Common Pitfalls & Exam Traps
Pitfall 1 (Position n must be natural): Position n must be a positive integer (1, 2, 3, …). If solving tn = value yields a fraction, that value does not belong to the sequence.
Pitfall 2 (GP Index Confusion): Always use (n-1) in the exponent: tn = a · rn-1. For the first term (n=1), r⁰ = 1.