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Number Sequences Guide

Arithmetic Sequences

Arithmetic sequence: constant difference d between consecutive terms. nth term: aₙ = a + (n-1)d. Where a = first term, d = common difference. Sum of n terms: Sₙ = n/2 × (2a + (n-1)d) = n/2 × (first term + last term). Example: 3, 7, 11, 15... d=4, a=3. nth term: 3 + (n-1)×4 = 4n - 1. 20th term: 4×20 - 1 = 79. Sum to 20: 20/2 × (3+79) = 10×82 = 820. Arithmetic sequences describe: salary increases by a fixed amount each year. Equally spaced data points. Simple interest accumulation.

Geometric Sequences

Geometric sequence: constant ratio r between consecutive terms. nth term: aₙ = ar^(n-1). Sum of n terms: Sₙ = a(1-rⁿ)/(1-r) for r≠1. Infinite sum (|r| < 1): S∞ = a/(1-r). Example: 2, 6, 18, 54... r=3, a=2. nth term: 2×3^(n-1). 10th term: 2×3⁹ = 2×19683 = 39,366. Sum to 10: 2×(1-3¹⁰)/(1-3) = 2×(-59048)/(-2) = 59,048. Geometric sequences describe: compound interest. Exponential population growth. Radioactive decay. Computer algorithm complexity O(2ⁿ).

Quadratic Sequences

Quadratic sequence: constant second difference. First differences vary, but the differences of the differences are constant. nth term: aₙ = an² + bn + c. Finding the formula: second difference = 2a → find a. Substitute two known terms to find b and c. Example: 1, 4, 9, 16, 25... First differences: 3, 5, 7, 9... Second differences: 2, 2, 2... (constant). Second difference = 2a = 2 → a=1. So n² fits exactly. General quadratic: when second difference is k, the n² coefficient is k/2.

Fibonacci and Special Sequences

Fibonacci: 1, 1, 2, 3, 5, 8, 13, 21, 34... Each term = sum of previous two. Golden ratio: limit of consecutive Fibonacci ratios → φ = (1+√5)/2 ≈ 1.618. Appears in: plant spiral patterns (phyllotaxis), shell spirals, art and architecture proportions. Triangular numbers: 1, 3, 6, 10, 15... nth term: n(n+1)/2. Sum of first n natural numbers. Square numbers: 1, 4, 9, 16... nth term: n². Cube numbers: 1, 8, 27, 64... nth term: n³. Prime numbers: 2, 3, 5, 7, 11, 13... No closed-form nth term formula e