How this calculator works
Choose a defined family and starting index to generate consecutive terms. Arithmetic and geometric families use an entered value at index zero. Standard Fibonacci numbers also start at index zero. Prime and star number lists begin at index one. Integer-defined families use exact integers, while arithmetic and geometric entries use floating-point numbers. This tool generates known families; it does not infer an arbitrary rule from a short list.
Formula
Arithmetic: aₙ=a₀+nd. Geometric: aₙ=a₀rⁿ. Triangular: n(n+1)/2. Square: n². Cube: n³. Star: 6n(n-1)+1. Factorial: n!=1×...×n, with 0!=1.
Worked example
Triangular indices 1 through 5 give 1, 3, 6, 10, 15. Prime indices 3 through 5 give 5, 7, 11. Standard Fibonacci indices 0 through 4 give 0, 1, 1, 2, 3.
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Frequently asked questions
Can it predict the next term of any list?
No. A finite list can fit infinitely many rules. Select a mathematical family whose definition matches your intended sequence.
Why do indices differ by family?
Zero is useful for powers, factorials and Fibonacci seeds. The first prime and first star number are conventionally labelled index one here.
Are all displayed values exact?
Integer-defined families are exact. Arithmetic and geometric sequences use floating-point arithmetic, so decimal inputs and large powers may be approximate.