Sequence Calculator (Arithmetic/Geometric)

A sequence is an ordered list of numbers following a pattern. The two most common types are arithmetic (constant difference) and geometric (constant ratio). Arithmetic sequence: Each term increases by a fixed amount (common difference d). 3, 7, 11, 15... (d=4). nth term: a(n) = a₁ + (n-1)d. Sum of n terms: S = n/2 × (first + last). Geometric sequence: Each term is multiplied by a fixed ratio (r). 2, 6, 18, 54... (r=3). nth term: a(n) = a₁ × r^(n-1). Sum: S = a₁(1-rⁿ)/(1-r) when r≠1. Real applications: Arithmetic: salary raises, equally-spaced payments, temperature changes. Geometric: compound interest (money grows geometrically), population growth, radioactive decay, viral spread. Infinite geometric series: If |r| < 1, the sum converges to a finite value: S = a₁/(1-r). Example: 1 + 0.5 + 0.25 + 0.125... = 1/(1-0.5) = 2.