Number Sequence Calculator

Sequence problems on tests look easy until you are staring at 3, 7, 11, 15… and need the 20th term plus the partial sum—without mis-counting steps on scrap paper. This page splits into Identify (paste terms, get the rule) and Generate (pick arithmetic or geometric and build forward).

How to Use This Number Sequence Calculator

• Identify Sequence tab. Paste at least three terms separated by commas or spaces. Enter target term index n. Press Analyze to see type, common difference or ratio, formula, next five terms, a_n, and S_n.
• Generate Sequence tab. Choose arithmetic or geometric, enter first term a₁, difference d or ratio r, and target n. Press Generate for a_n, S_n, and the first ten terms.
• Read error messages honestly. Patterns outside arithmetic, geometric, or Fibonacci-like rules report as unknown— the tool will not force-fit a wrong formula.
• Pair with other tools. For general list statistics on the terms you found, use the Mean Median Mode Range Calculator. For exponent-heavy growth checks, open the Exponent Calculator.

Sequence Formulas and Practical Applications

Arithmetic sequences step by addition; geometric sequences step by multiplication. Fibonacci-like lists add the previous two values—useful when growth depends on two prior states, not a fixed increment.

Arithmetic sequences

a_n = a₁ + (n − 1)d

S_n = n(a₁ + a_n) / 2

Constant difference d means each hop adds the same amount—think evenly spaced fence posts or hourly wage accrual. For 3, 7, 11, 15, d = 4 and the 10th term is 3 + 9×4 = 39.

Geometric sequences

a_n = a₁ × r^(n − 1)

S_n = a₁(1 − r^n) / (1 − r) when r ≠ 1

Ratio r multiplies each term—compound interest without deposits behaves geometrically. When r = 1, every term equals a₁ and S_n = n×a₁.

Fibonacci-like detection

a_n = a_{n−1} + a_{n−2}

The Identify tab checks whether each term after the second equals the sum of its two predecessors. Classic Fibonacci starts 1, 1, 2, 3, 5… but any seed pair defines its own Fibonacci-like chain. Very large n (> 1000) caps a_n and S_n to keep runtime sane.

Frequently Asked Questions

What sequence types can the Identify tab detect?

Arithmetic, geometric, and Fibonacci-like—in that priority order. Custom quadratic or polynomial patterns are not auto-detected.

How many terms are needed to identify a sequence?

Minimum three. Two terms always fit infinitely many rules; three constraints usually pin down d or r.

What is a Fibonacci-like sequence?

Each term after the second is the sum of the two before it, regardless of starting pair. The formula row shows a_n = a_{n−1} + a_{n−2}.

Can common difference or ratio be negative?

Yes—negative d counts down; negative r flips sign each step (e.g., 2, −6, 18, −54 with r = −3).

What does the Generate tab do?

Skips detection and builds from your a₁ and d or r directly—handy when the rule is already known. Browse math calculators for related tools.