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Number Sequence Calculator
Identify an existing number sequence and find its general rule, or generate a custom arithmetic or geometric sequence.
Analysis Results
| Sequence Type | — |
|---|---|
| Common Difference / Ratio | — |
| General Formula | — |
| Next 5 Terms | — |
| Target Term (a_n) | — |
| Sum of n Terms (S_n) | — |
Generation Results
| Target Term (a_n) | — |
|---|---|
| Sum of n Terms (S_n) | — |
| Generated Terms | — |
Arithmetic vs. Geometric Sequences
A number sequence is an ordered list of numbers that follow a specific pattern.
An arithmetic sequence is one where each term is found by adding a constant value (the common difference, d) to the previous term.
Formula for the n-th term: a_n = a_1 + (n - 1)d
Formula for the sum of the first n terms: S_n = n(a_1 + a_n) / 2
A geometric sequence is one where each term is found by multiplying the previous term by a constant value (the common ratio, r).
Formula for the n-th term: a_n = a_1 * r^(n - 1)
Formula for the sum of the first n terms (where r ≠ 1): S_n = a_1(1 - r^n) / (1 - r). If r = 1, then S_n = n * a_1.
How to Identify a Sequence
To identify a sequence, look at the difference and ratio of consecutive terms:
- If a_2 - a_1 = a_3 - a_2, the sequence is arithmetic. The common difference is this constant value.
- If a_2 / a_1 = a_3 / a_2, the sequence is geometric. The common ratio is this constant value.
- If each term is the sum of the preceding two terms (a_n = a_{n-1} + a_{n-2}), it is a Fibonacci-like sequence.
FAQ
What is a Fibonacci sequence?
A Fibonacci sequence starts with two values (typically 0 and 1 or 1 and 1) where every subsequent term is the sum of the preceding two terms: 1, 1, 2, 3, 5, 8, 13, 21, etc.
Can a sequence have a negative common difference or ratio?
Yes. A negative common difference means the sequence decreases (e.g. 10, 7, 4, 1...). A negative common ratio means the terms alternate in sign (e.g. 2, -6, 18, -54...).