Sequences, Series & Mathematical Induction
Sequences are ordered lists of numbers following a pattern. Series sum them up. Together they model everything from loan payments to the behavior of algorithms.
Key Concepts
Arithmetic Sequences
Each term increases by a constant difference d. aₙ = a₁ + (n−1)d. Sum of first n terms: Sₙ = n(a₁+aₙ)/2 = n[2a₁+(n−1)d]/2. Example: 2, 5, 8, 11, ... (d=3).
Geometric Sequences
Each term is multiplied by a constant ratio r. aₙ = a₁·rⁿ⁻¹. Sum: Sₙ = a₁(1−rⁿ)/(1−r). Infinite geometric series: S∞ = a₁/(1−r) when |r| < 1.
Sigma Notation
∑ᵢ₌₁ⁿ f(i) means sum f(1)+f(2)+...+f(n). ∑ᵢ₌₁ⁿ i = n(n+1)/2. ∑ᵢ₌₁ⁿ i² = n(n+1)(2n+1)/6. Useful for expressing repeated addition compactly.
Mathematical Induction
Prove P(n) is true for all n≥1: 1) Base case: prove P(1). 2) Inductive step: assume P(k) is true, prove P(k+1). Like dominoes falling — knock the first, then show each knocks the next.
Live Python Practice
Interactive Lab
Check Your Understanding
In arithmetic sequence 3, 7, 11, 15, ..., the 10th term is:
The sum S∞ of a geometric series with a₁=4 and r=0.5 is:
Mathematical induction requires: