Number Sequence Calculator

Generate arithmetic or geometric sequences. Find the nth term and sum of a series given the first term and common difference or ratio.

The first number in the sequence.

For arithmetic sequences. Set to 0 if using geometric.

For geometric sequences. Set to 0 if using arithmetic.

How many terms to generate.

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Arithmetic vs Geometric Sequences

An arithmetic sequence adds a constant difference between terms (2, 5, 8, 11 — difference of 3). A geometric sequence multiplies by a constant ratio (2, 6, 18, 54 — ratio of 3). This calculator generates either type and computes the sum of all terms.

Arithmetic Sequence Formulas

The nth term: aₙ = a₁ + (n-1)d. The sum of n terms: Sₙ = n/2 × (2a₁ + (n-1)d) or Sₙ = n/2 × (a₁ + aₙ). These formulas allow you to find any term or the total sum without listing all terms.

Geometric Sequence Formulas

The nth term: aₙ = a₁ × r^(n-1). The sum of n terms: Sₙ = a₁ × (1 - rⁿ) / (1 - r) when r ≠ 1. For infinite geometric series with |r| < 1, the sum converges to S = a₁ / (1 - r). This formula underlies compound interest and present value calculations.

Sequences in Mathematics and Science

Arithmetic sequences model linear growth (constant salary increases). Geometric sequences model exponential growth (compound interest, population growth) and decay (depreciation). The Fibonacci sequence, while neither arithmetic nor geometric, appears throughout nature in spiral patterns and plant growth.

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