Number Sequence Calculator
Arithmetic sequence
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How This Calculator Detects a Pattern
Enter at least three known terms of a sequence, in order, separated by commas. The calculator first checks whether the sequence is arithmetic — each term formed by adding a constant common difference d to the previous one, so that a(n) = a(1) + (n - 1) × d. If the differences between consecutive terms aren't constant, it checks whether the sequence is geometric instead — each term formed by multiplying the previous one by a constant common ratio r, so that a(n) = a(1) × r^(n - 1). These are the two most common well-defined sequence types, and together with the constant/repeating case they cover the overwhelming majority of number sequences you'll encounter in algebra courses and everyday problems.
Sum of the Sequence
Once a pattern is confirmed, the sum of the known and predicted terms is computed with the standard closed-form sums rather than by brute-force addition: for an arithmetic sequence, the sum of the first n terms is S(n) = n/2 × (a(1) + a(n)); for a geometric sequence with ratio r ≠ 1, it's S(n) = a(1) × (1 - r^n) / (1 - r). These formulas are exact and avoid the rounding drift that can creep in from repeated multiplication over many terms.
When the Pattern Isn't Arithmetic or Geometric
Some sequences — like the Fibonacci sequence, where each term is the sum of the two before it, or sequences of perfect squares and cubes — follow a rule that isn't a constant difference or ratio. If this calculator can't confirm an arithmetic or geometric pattern from your terms, double-check the order and count of the values you entered; a minimum of three or four consecutive terms is usually needed to confirm a pattern with confidence. For general work with the individual numbers in a sequence, such as sums, averages, or spread, the statistics calculator and average calculator can help you analyze the values directly.
Frequently Asked Questions
How many terms do I need to enter for accurate pattern detection?
Enter at least 3 terms, though 4 or more is safer. With only 3 terms, a coincidental constant difference or ratio could be detected that doesn't reflect the sequence's true underlying rule, especially with rounded or noisy data. More consecutive terms give the calculator more checks to confirm the pattern holds consistently.
What if my sequence is neither arithmetic nor geometric, like Fibonacci?
This calculator only detects arithmetic (constant difference) and geometric (constant ratio) patterns, since those have universally agreed, closed-form definitions. Sequences like Fibonacci (each term is the sum of the previous two) or perfect squares follow other rules and will show as 'no pattern detected' here - you would need to compute those terms with their own specific recurrence relation.