Find patterns in numbers and predict what comes next. Automatically detects various patterns including arithmetic, geometric, and Fibonacci sequences.
A sequence is a set of numbers arranged in a specific order following a rule. Arithmetic sequences add a constant, geometric sequences multiply by a constant, Fibonacci sequences add the previous two terms. Understanding patterns allows predicting next terms and deriving general formulas.
Arithmetic sequences have constant differences between consecutive terms. Example: 2, 5, 8, 11, 14... with common difference 3. General term: aₙ = a₁ + (n-1)d. Sum formulas: Sₙ = n(a₁ + aₙ)/2 or Sₙ = n[2a₁ + (n-1)d]/2.
Geometric sequences have constant ratios between consecutive terms. Example: 3, 6, 12, 24, 48... with common ratio 2. General term: aₙ = a₁ × r^(n-1). Sum: Sₙ = a₁(1-rⁿ)/(1-r). Used in compound interest, population growth predictions.
Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21... defined as F(n) = F(n-1) + F(n-2). Ratio of consecutive terms approaches golden ratio (≈1.618). Found in nature's spiral patterns, flower petal arrangements, stock market analysis (Fibonacci retracement).
Square numbers (1, 4, 9, 16, 25...), cube numbers (1, 8, 27, 64...), prime numbers (2, 3, 5, 7, 11...), triangular numbers (1, 3, 6, 10, 15...). Each has unique mathematical properties, applied in cryptography, algorithm optimization.
Bank compound interest (geometric), building stair design (arithmetic), cell division (geometric), programming recursion (Fibonacci), physics uniform acceleration (arithmetic), economics growth rate prediction. Sequences have practical applications across many fields.