Identify, analyze, and generate arithmetic, geometric, Fibonacci, and quadratic number sequences. Find next terms, explicit formulas, and partial sums.
The difference between consecutive terms is constant, confirming an Arithmetic Progression.
Direct evaluation using the arithmetic sequence formula.
Calculated using Gauss summation.
A number sequence is an ordered set of numbers governed by a mathematical pattern or rule. The two most fundamental sequences are Arithmetic Progressions (where each term adds a constant difference d) and Geometric Progressions (where each term multiplies by a constant ratio r).
A sequence is arithmetic if and only if the first difference a_{k+1} - a_k is constant for all consecutive terms.
A sequence is geometric if the ratio a_{k+1} / a_k is identical across all non-zero terms.
F_n = F_{n-1} + F_{n-2}. The limit of ratios F_{n+1} / F_n equals the golden ratio φ = (1 + √5) / 2 ≈ 1.6180339887.
Find the 20th term and sum of the first 20 terms of: 5, 8, 11, 14, 17...
An explicit formula allows you to calculate any term a_n directly without computing intermediate terms (e.g. a_n = 2n + 1). A recursive formula defines each term relative to its predecessor (e.g. a_n = a_{n-1} + 2).
An infinite geometric series converges to a finite sum if and only if the absolute value of the common ratio is strictly less than 1 (|r| < 1). If |r| ≥ 1, the series diverges.
Understanding your calculation
Sequence tools examine patterns such as a constant difference or constant ratio and use the selected rule to extend the terms.
2, 5, 8, 11 is arithmetic with difference 3, so its next term under that rule is 14.
A finite list can fit many different rules. A suggested continuation is not proof that the original sequence has that unique pattern.