Number Sequence Calculator

Number sequences are among the oldest objects of mathematical study — the ancient Greeks analyzed arithmetic progressions and geometric progressions, Fibonacci described his rabbit sequence in 1202, and sequences remain central to modern mathematics, computer science, and data analysis. The ability to identify a sequence's pattern, predict its next term, and find formulas for arbitrary terms is a core mathematical reasoning skill that transfers to modeling population growth, calculating interest, analyzing algorithms, and identifying patterns in data. Understanding the main families of sequences and how to work with each builds the pattern recognition that drives mathematical intuition.

Identifying Unknown Sequences

When presented with an unknown sequence, systematic checking reveals the pattern. First, calculate first differences (subtract consecutive terms). If constant, arithmetic. If not constant, calculate second differences (differences of differences). If constant, quadratic (polynomial of degree 2). Calculate third differences; if constant, cubic. If first differences form a geometric sequence, the original may be geometric-like.

Sequence: 2, 6, 14, 26, 42, 62, ... First differences: 4, 8, 12, 16, 20 (not constant). Second differences: 4, 4, 4, 4 (constant!). Second differences are constant → the sequence is quadratic. A quadratic sequence has formula aₙ = An² + Bn + C. The second difference equals 2A: 4 = 2A → A = 2. Using a₁ = 2: 2(1) + B + C = 2 → B + C = 0. Using a₂ = 6: 2(4) + 2B + C = 6 → 8 + 2B + C = 6 → 2B + C = -2. Solve: B = -2, C = 2. Formula: aₙ = 2n² - 2n + 2. Verify: a₃ = 2(9)-6+2 = 14. Correct.

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Arithmetic Sequences

An arithmetic sequence increases (or decreases) by a constant difference d between consecutive terms. General form: a, a+d, a+2d, a+3d, ... The nth term formula: aₙ = a₁ + (n-1)d, where a₁ is the first term and d is the common difference.

Sequence: 7, 11, 15, 19, 23, ... Difference d = 4 (constant). First term a₁ = 7. Formula: aₙ = 7 + (n-1) × 4 = 7 + 4n - 4 = 4n + 3. Verify: a₁ = 4(1)+3 = 7. a₅ = 4(5)+3 = 23. Both correct. The 50th term: a₅₀ = 4(50)+3 = 203.

Sum of first n terms (arithmetic series): Sₙ = n/2 × (a₁ + aₙ) = n/2 × (2a₁ + (n-1)d). Sum of the first 50 terms of 7, 11, 15, ...: S₅₀ = 50/2 × (7 + 203) = 25 × 210 = 5,250. This formula (and Gauss's childhood insight that pairs of terms from the beginning and end of a sequence sum to the same value) is one of the beautiful results of elementary mathematics.

Special Sequences

Square numbers: 1, 4, 9, 16, 25, 36 → aₙ = n². Cube numbers: 1, 8, 27, 64, 125 → aₙ = n³. Triangular numbers: 1, 3, 6, 10, 15, 21 → aₙ = n(n+1)/2. Pentagonal numbers: 1, 5, 12, 22, 35 → aₙ = n(3n-1)/2. Powers of 2: 1, 2, 4, 8, 16, 32 → aₙ = 2^(n-1). Primes: 2, 3, 5, 7, 11, 13, 17 → no closed-form formula; identified by trial division or sieve.

Recognizing these sequences by sight is a useful skill in competitive mathematics and algorithm analysis. When a runtime or space complexity involves triangular numbers (operations scale as n(n+1)/2 ≈ n²/2), recognizing the pattern immediately tells you the algorithm is O(n²).

Recursive vs Closed-Form Formulas

Many sequences are defined recursively (each term depends on previous terms) but can be expressed in closed form (a direct formula for the nth term). Fibonacci numbers: defined recursively as F(n) = F(n-1) + F(n-2), expressed in closed form by Binet's formula. Arithmetic sequences: defined recursively as aₙ = aₙ₋₁ + d, expressed in closed form as aₙ = a₁ + (n-1)d.

Closed-form expressions are generally preferable for computation because they compute any term in O(1) time rather than O(n) time for recursive evaluation. The Tower of Hanoi sequence — number of moves to solve n disks: 1, 3, 7, 15, 31 — has closed form aₙ = 2ⁿ - 1. Knowing this immediately answers "how many moves does a 64-disk puzzle require?" (2^64 - 1 = 18,446,744,073,709,551,615 moves, which at 1 move per second would take about 585 billion years).

Geometric Sequences

A geometric sequence multiplies each term by a constant ratio r to get the next. General form: a, ar, ar², ar³, ... The nth term: aₙ = a₁ × r^(n-1).

Sequence: 3, 6, 12, 24, 48, ... Ratio r = 2. First term a₁ = 3. Formula: aₙ = 3 × 2^(n-1). The 10th term: a₁₀ = 3 × 2⁹ = 3 × 512 = 1,536. Sum of geometric series: Sₙ = a₁ × (r^n - 1)/(r - 1) for r ≠ 1. Sum of first 10 terms: S₁₀ = 3 × (2^10 - 1)/(2-1) = 3 × 1,023 = 3,069.

Exponential growth and decay in real world phenomena are geometric sequences when measured at discrete time intervals. Population growing at 3% per year: after year 0 (baseline P₀), year n has population P₀ × 1.03^n. Radioactive decay at 12% per year: remaining material after n years = A₀ × 0.88^n. Compound interest is a geometric sequence: balance after n years at rate r = P₀ × (1+r)^n.

Sequences in Programming and Algorithm Analysis

Algorithm complexity is expressed as a function of input size n. Understanding whether a function grows arithmetically (linear O(n)), quadratically (O(n²)), geometrically (O(2ⁿ)), or logarithmically (O(log n)) determines practical performance at large scale. An algorithm that takes 1 second for n=1,000 takes about 1,000 seconds (16 minutes) for n=1,000,000 if linear, but 1,000,000,000 seconds (31 years) if quadratic, and only about 2 seconds if logarithmic.

The OEIS (Online Encyclopedia of Integer Sequences) catalogs over 370,000 integer sequences, providing closed-form formulas, generating functions, references, and cross-links. When you encounter a sequence in research or problem-solving and can't identify the pattern, entering the first several terms into OEIS often immediately identifies it and provides complete mathematical information. It's the sequence-finder used by mathematicians worldwide — and freely available at oeis.org.