📐 Sequence Calculator

Arithmetic sequences add a constant value (common difference) to each term, resulting in linear growth. Geometric sequences multiply each term by a constant value (common ratio), resulting in exponential growth or decay. For example, 2, 4, 6, 8 is arithmetic (adding 2), while 2, 4, 8, 16 is geometric (multiplying by 2).

For arithmetic sequences, use aₙ = a₁ + (n-1)d. For geometric sequences, use aₙ = a₁ × r^(n-1). For Fibonacci, you need to calculate all previous terms or use Binet's formula. This calculator automatically computes the nth term for you based on your inputs.

The sum of a sequence (also called a series) is the total when you add all terms together. For arithmetic sequences, use Sₙ = n/2 × (a₁ + aₙ). For geometric sequences, use Sₙ = a₁ × (1 - r^n) / (1 - r) when r ≠ 1. The calculator displays both the nth term and the sum of all terms.

The Fibonacci sequence appears throughout nature in flower petals, spiral shells, tree branches, and more. The ratio between consecutive Fibonacci numbers approaches the golden ratio (φ ≈ 1.618), which is considered aesthetically pleasing and appears in art and architecture. It's also fundamental in computer science algorithms.

Yes! Arithmetic sequences can have negative common differences (decreasing sequences) or negative first terms. Geometric sequences can have negative ratios (alternating positive/negative terms) or negative first terms. The calculator handles all these cases correctly.

When r = 1, all terms in the geometric sequence are equal to the first term. The sum formula simplifies to Sₙ = n × a₁. This is actually a special case of an arithmetic sequence with d = 0. The calculator handles this case automatically.

Arithmetic sequences model regular savings or payment plans (like monthly deposits). Geometric sequences model compound interest and investment growth. For example, if you invest $1000 at 5% annual interest, your balance each year forms a geometric sequence with ratio 1.05.