CALTRX

Number Sequence Calculator

Find the nth term and sum of arithmetic or geometric sequences. Shows the formula, pattern, and full working for any sequence.

Enter your values above to see the results.

Tips & Notes

  • ✓Identify type: check differences (arithmetic) or ratios (geometric). Constant difference → arithmetic.
  • ✓Arithmetic nth term: start + (n−1)×difference. 100th term of 3,7,11...: 3+99×4=399.
  • ✓Sum of 1+2+3...+n = n(n+1)/2. Sum of 1+3+5...+(2n−1) = n². Classic results.
  • ✓Geometric sum shortcut: double all terms minus first. For 1+2+4+8: 2(1+2+4+8)−1=31.
  • ✓Test geometric: if a₂/a₁ = a₃/a₂ = a₄/a₃ = r, it is geometric with ratio r.

Common Mistakes

  • ✗Arithmetic vs geometric confusion: 2,4,8,16 is geometric (ratio 2), not arithmetic (differences vary).
  • ✗Off-by-one in nth term: 10th term of 3,7,11... is a₁+9d=3+36=39, not 3+10×4=43.
  • ✗Geometric sum formula: denominator is (1−r), not (r−1). Check sign carefully.
  • ✗Infinite geometric sum only converges for |r|<1. 1+2+4+8... diverges.
  • ✗Assuming a pattern with only 2-3 terms — need at least 3-4 terms to identify type confidently.

Number Sequence Calculator Overview

A number sequence is an ordered list of numbers following a definable pattern. Identifying the type of sequence — and its governing rule — allows prediction of any term, calculation of the sum of any number of terms, and detection of underlying mathematical structure. From simple counting numbers (1,2,3,4...) to complex recursive sequences like Fibonacci, sequences are the discrete counterpart to continuous functions and appear throughout finance (payment schedules), physics (quantum energy levels), and computer science (algorithm analysis).

Arithmetic Sequence — each term increases by a constant difference d:

aₙ = a₁ + (n−1)d
EX: 3, 7, 11, 15, 19... → d=4, a₁=3 → a₁₀ = 3+9×4 = 39 | Sum of first 10 = 10/2×(3+39) = 210
Geometric Sequence — each term multiplied by constant ratio r:
aₙ = a₁ × r^(n−1)
EX: 2, 6, 18, 54... → r=3, a₁=2 → a₇ = 2×3⁶ = 1,458 | Sum S₅ = 2×(1−3⁵)/(1−3) = 242
Infinite geometric series — converges to a finite sum only when |r| < 1:
EX: 1+0.5+0.25+... = 1/(1−0.5) = 2 | EX: 0.333... = (1/3)/(1−1/10) ... = 1/3 exactly
Identifying sequence type: - Constant differences → Arithmetic (linear growth) - Constant ratios → Geometric (exponential growth or decay) - Constant second differences → Quadratic (polynomial growth) - Each term = sum of two previous → Fibonacci-type (exponential with φ base) Sum formulas:
Arithmetic: Sₙ = n/2 × (2a₁+(n−1)d) | Sum 1 to 100: n=100, a₁=1, d=1 → 100/2×(2+99) = 5,050
Geometric: Sₙ = a₁×(1−rⁿ)/(1−r) | 1+2+4+8+16+32 (a₁=1,r=2,n=6): (1−64)/(1−2) = 63
Recognizing which type of sequence you are dealing with determines the entire solution approach. Arithmetic sequences (constant difference) and geometric sequences (constant ratio) have closed-form formulas for any term and any sum. Fibonacci and other recursive sequences require knowing multiple previous terms. When a sequence is neither arithmetic nor geometric, look for a difference sequence: compute differences between consecutive terms — if those differences are constant, the original is arithmetic of degree 2, solvable with quadratic formulas. Sequences appear throughout applied mathematics in ways that are not always obvious. Loan amortization creates an arithmetic structure in payment counts. Population growth at a constant percentage rate is geometric. Algorithm efficiency classes — O(n), O(n²), O(2ⁿ) — correspond to arithmetic, polynomial, and geometric sequences in computational cost. Recognizing the sequence type behind a real-world problem reduces it to a standard formula rather than requiring case-by-case numerical computation.

Frequently Asked Questions

An arithmetic sequence adds a constant difference d each term. Identify by subtracting consecutive terms — if all differences are equal, it is arithmetic. Formula: a(n) = a(1) + (n−1)×d. Example: 3, 7, 11, 15, 19... Differences: 4, 4, 4, 4 (constant). d = 4, a(1) = 3. Formula: a(n) = 3 + (n−1)×4 = 4n − 1. Term 10: a(10) = 3 + 9×4 = 39. Sum of first n terms: S(n) = n/2 × (first + last) = n/2 × (2a₁ + (n−1)d).

A geometric sequence multiplies by a constant ratio r each term. Identify by dividing consecutive terms — if all ratios are equal, it is geometric. Formula: a(n) = a(1) × r^(n−1). Example: 2, 6, 18, 54, 162... Ratios: 3, 3, 3, 3. r = 3, a(1) = 2. Formula: a(n) = 2 × 3^(n−1). Term 8: a(8) = 2 × 3⁷ = 2 × 2187 = 4374. Sum of first n terms: S(n) = a(1) × (rⁿ − 1)/(r − 1). For |r| < 1, the infinite sum = a(1)/(1−r).

A Fibonacci-like sequence adds the two previous terms: a(n) = a(n−1) + a(n−2). Starting values determine the sequence. Classic Fibonacci: 1, 1, 2, 3, 5, 8, 13, 21... Lucas sequence: 2, 1, 3, 4, 7, 11, 18... Any Fibonacci-like sequence eventually has consecutive term ratios converging to the golden ratio φ ≈ 1.618, regardless of starting values. To identify: check if each term equals the sum of the two before it.

Square numbers: 1, 4, 9, 16, 25, 36... (n²). Cubic numbers: 1, 8, 27, 64, 125... (n³). Triangular numbers: 1, 3, 6, 10, 15, 21... (n(n+1)/2). Powers of 2: 1, 2, 4, 8, 16, 32... (2^(n−1)). Factorial sequence: 1, 1, 2, 6, 24, 120... (n!). These sequences appear throughout mathematics — triangular numbers are sums of arithmetic series, square numbers appear in Pythagorean triples, and powers of 2 define binary counting and computer memory sizes.

Find differences between consecutive terms. If differences are constant: arithmetic. If differences of differences are constant: polynomial of that degree. Example: 1, 4, 9, 16, 25... First differences: 3, 5, 7, 9 (not constant). Second differences: 2, 2, 2 (constant) → quadratic sequence (n²). If ratios between terms are constant: geometric. If none of these patterns fit, try looking for Fibonacci-type or prime number patterns. The OEIS (Online Encyclopedia of Integer Sequences) catalogs hundreds of thousands of known sequences.

Start by computing first differences (subtract each term from the next). If constant → arithmetic sequence. If not constant, compute second differences. If constant → quadratic sequence (nth term is a quadratic formula). If second differences are also not constant, compute third differences. Constant third differences → cubic sequence. For sequences like 1, 2, 4, 8, 16: compute ratios instead (each term divided by previous). Constant ratio → geometric sequence. For sequences that do not fit arithmetic or geometric patterns (like 1, 1, 2, 3, 5, 8), look for Fibonacci-type patterns where each term is the sum of previous terms. This calculator supports arithmetic, geometric, and Fibonacci sequences.