Number Sequence Calculator
Generate terms, find the nth term, and calculate partial sums for arithmetic and geometric sequences. Paste your own numbers and we'll identify the pattern.
aₙ
Nth Term
Sₙ
Partial Sum
a₁·r,+d
Pattern Type
Arithmetic Sequence
aₙ = a₁ + (n-1)×d Sₙ = n/2 × (2a₁ + (n-1)d)🔍 Sequence Identifier — Paste Numbers to Detect Type
Enter a comma or space-separated sequence (at least 3 terms) and we'll identify what type it is.
Detected: arithmetic
Common difference d = 3
aₙ = a₁ + (n-1)×3
Next terms: 17, 20, 23, 26, 29
First 15 Terms (Term #10 highlighted)
2, 5, 8, 11, 14, 17, 20, 23, 26, 29, 32, 35, 38, 41, 44
29
Term #10 (nth)
155
Sum of 10 Terms
345
Sum of All 15
2
First Term
44
Last Term
15
Total Terms
2
Min
44
Max
📈 Sequence Visualization
Arithmetic vs. Geometric Sequences
A number sequence is an ordered list of numbers that follows a rule. An arithmetic sequence adds (or subtracts) the same amount every step — the common difference, d. For example, 2, 5, 8, 11, 14 is arithmetic with d = 3, since every term is 3 more than the one before it. A geometric sequence instead multiplies by the same amount every step — the common ratio, r. For example, 3, 6, 12, 24, 48 is geometric with r = 2, since every term is double the one before it. Paste any list of numbers into the identifier above and this calculator checks the differences and ratios for you.
The Key Formulas
Arithmetic nth term: aₙ = a₁ + (n-1)×dArithmetic sum: Sₙ = n/2 × (2a₁ + (n-1)d)Geometric nth term: aₙ = a₁ × r^(n-1)Geometric sum: Sₙ = a₁(1-rⁿ)/(1-r), for r ≠ 1Geometric sum to infinity (|r| < 1): S = a₁/(1-r)A quick note on quadratic sequences: some sequences aren't arithmetic or geometric but still follow a rule — if the differences between terms themselves form an arithmetic sequence (a constant “second difference”), the sequence is quadratic. Our identifier checks for this pattern too once the simpler arithmetic and geometric tests are ruled out.
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