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Arithmetic, geometric and Fibonacci sequences with nth term formula, sum formula and first 10 terms
The Number Sequence Calculator analyzes arithmetic, geometric, and Fibonacci sequences. Given any starting conditions, it derives the nth term formula, computes the partial sum formula, and lists the first 10 (or more) terms. Students studying sequences and series, as well as anyone working with recurring patterns in data, use this to understand how a sequence behaves and grows over time.
Frequently Asked Questions
Sobre Esta Calculadora
Number sequences are ordered lists of numbers following a defined pattern or rule. The three classic types are: arithmetic sequences (constant difference between consecutive terms: 2, 5, 8, 11…), geometric sequences (constant ratio between consecutive terms: 3, 6, 12, 24…), and the Fibonacci sequence (each term is the sum of the two preceding terms: 1, 1, 2, 3, 5, 8…).
For arithmetic sequences, the nth term is aₙ = a₁ + (n−1)d and the sum of the first n terms is Sₙ = n/2 × (a₁ + aₙ). For geometric sequences, the nth term is aₙ = a₁ × rⁿ⁻¹ and the sum is Sₙ = a₁(rⁿ − 1)/(r − 1). When |r| < 1, the geometric series converges to a finite sum as n → ∞.
Sequences appear throughout mathematics (Taylor series), finance (compound interest is a geometric sequence of balances), biology (population growth), computer science (recurrence relations in algorithm analysis), and nature (Fibonacci patterns in plant growth and spiral shells).
Cálculo lógico.
Exemplo Resolvido
Analyzing the geometric sequence starting at 3 with ratio 4
Entradas:
- Sequence type: Geometric
- First term (a₁): 3
- Common ratio (r): 4
Passo a Passo:
- nth term formula: aₙ = a₁ × r^(n−1) = 3 × 4^(n−1)
- First 6 terms: a₁=3, a₂=12, a₃=48, a₄=192, a₅=768, a₆=3072
- Sum of first n terms: Sₙ = a₁(rⁿ − 1)/(r − 1) = 3(4ⁿ − 1)/3 = 4ⁿ − 1
- Sum of first 5 terms: S₅ = 3(4⁵ − 1)/(4 − 1) = 3(1024 − 1)/3 = 1023