Enter at least 3 terms of a sequence to detect whether it is arithmetic or geometric, and find any term you need.
A Number Sequence Calculator identifies the pattern in a series of numbers and predicts subsequent terms, or generates a sequence based on a specified rule, such as arithmetic (constant difference), geometric (constant ratio), or Fibonacci-style (each term based on previous terms). This tool is useful for solving pattern recognition problems, exploring number theory, and checking sequence-related homework or puzzles.
For an arithmetic sequence, each term equals the previous term plus a constant common difference: aₙ = a₁ + (n−1)d. For a geometric sequence, each term equals the previous term multiplied by a constant common ratio: aₙ = a₁ × r^(n−1). For a Fibonacci-style sequence, each term is the sum of the two preceding terms: aₙ = aₙ₋₁ + aₙ₋₂.
Enter a series of known numbers from your sequence, and the calculator attempts to identify the underlying pattern (arithmetic, geometric, or other common types) and predicts the next term or terms. Alternatively, specify a sequence type and starting parameters to generate a sequence directly.
Example 1: The sequence 3, 7, 11, 15 follows an arithmetic pattern with a common difference of 4, so the next term would be 15+4 = 19.
Example 2: The sequence 2, 6, 18, 54 follows a geometric pattern with a common ratio of 3, so the next term would be 54×3 = 162.
Check whether consecutive terms share a constant difference (arithmetic) by subtracting adjacent terms, or a constant ratio (geometric) by dividing adjacent terms; if either produces the same value consistently throughout the sequence, that identifies the pattern type.
The Fibonacci sequence (0, 1, 1, 2, 3, 5, 8, 13...) generates each term by summing the two before it, and remarkably, this pattern appears throughout nature in phenomena like flower petal arrangements, pinecone spirals, and shell growth patterns, making it one of the most famous mathematical sequences.
Arithmetic sequences grow (or shrink) at a constant, linear rate through consistent addition, while geometric sequences grow (or shrink) at an accelerating, exponential rate through consistent multiplication, leading to dramatically different long-term behavior between the two types.
Pattern recognition in number sequences tests logical reasoning ability and the capacity to identify underlying rules from limited examples, skills considered relevant to broader analytical and problem-solving capability, which is why they're a popular test format.
The sum of the first n terms of an arithmetic sequence uses the formula Sum = n/2 × (first term + last term), a shortcut derived from pairing terms from opposite ends of the sequence, which each sum to the same total.
The sum of the first n terms of a geometric sequence uses the formula Sum = a₁ × (1−r^n)/(1−r) (for r not equal to 1), a formula that accounts for the compounding multiplication effect characteristic of geometric growth.
Other notable sequences include square numbers (1, 4, 9, 16...), triangular numbers (1, 3, 6, 10...), and prime number sequences, each following its own distinct underlying mathematical rule beyond simple constant addition or multiplication.
Recognizing and generating sequences is foundational to algorithm design, particularly in areas like recursive programming (where Fibonacci-style sequences are a classic teaching example) and understanding computational complexity growth patterns.
With only a few known terms, more than one mathematical rule could technically fit the given numbers, which is why providing more terms in a sequence generally improves confidence in correctly identifying the true underlying pattern rather than a coincidental alternative.
A sequence is simply an ordered list of numbers following a pattern, while a series is the sum of the terms in a sequence, making series a natural extension of sequence concepts into cumulative totals rather than individual term values.
The declining balance of a loan over successive payments follows a mathematical sequence, with each period's balance depending on the previous one, making sequence concepts directly applicable to understanding how loan balances change over time.
Once the underlying rule connecting consecutive terms is identified, that same rule can be applied both forward and backward, making it possible to solve for a missing term anywhere within the sequence, not just at the very end.