Calculate arithmetic and geometric sequences — find any term, the sum of terms, and generate the full sequence with step-by-step solutions.
An arithmetic sequence is a list of numbers in which the difference between any two consecutive terms is constant. This constant is called the common difference (d). For example, 2, 5, 8, 11, 14 is arithmetic with d = 3. The nth term formula aₙ = a₁ + (n−1)d lets you find any term directly without listing all the terms before it.
A geometric sequence is a list of numbers in which each term is found by multiplying the previous term by a constant — the common ratio (r). For example, 3, 6, 12, 24, 48 is geometric with r = 2. When |r| < 1, the terms get progressively smaller and the infinite sum converges to a finite value: S∞ = a₁ ÷ (1−r).
The Fibonacci sequence starts with two seed values (classically 1 and 1) and each subsequent term is the sum of the two preceding terms: 1, 1, 2, 3, 5, 8, 13, 21, 34... It appears throughout nature in the arrangement of leaves, flower petals, spiral shells, and the branching of trees. The ratio of consecutive Fibonacci numbers converges to the golden ratio φ ≈ 1.618.
A sequence is an ordered list of numbers: 1, 2, 3, 4, 5. A series is the sum of the terms of a sequence: 1 + 2 + 3 + 4 + 5 = 15. This calculator computes both — the sequence (list of terms) and the series (sum of those terms).
Yes. A negative common ratio creates an alternating sequence — the terms alternate between positive and negative values. For example, with a₁ = 2 and r = −3: 2, −6, 18, −54, 162...
When |r| < 1 (e.g. r = 0.5), each term of a geometric series is smaller than the previous one, and the infinite sum approaches a finite limit. For example, 1 + 0.5 + 0.25 + 0.125 + ... = 2. When |r| ≥ 1, the series diverges (grows without bound).
The golden ratio φ ≈ 1.61803... is an irrational number with unique mathematical properties. As the Fibonacci sequence progresses, the ratio of consecutive terms (Fₙ₊₁ ÷ Fₙ) converges ever closer to φ. This relationship appears in art, architecture, and natural growth patterns.