Sequence Calculator
Stuck on a number sequence like 3, 7, 11, 15, … and need the 100th term or the sum of the first 50 terms? Maybe you’ve got a custom recurrence and want to avoid manual calculations. A sequence calculator gives you the answer in seconds.
A sequence is an ordered list of numbers following a specific rule. Our sequence calculator is a free online tool that finds missing terms, any particular term (the n‑th term), or the sum of a series for several common types–and can even infer the rule from a few given numbers.
Simply provide the first few terms of your sequence, choose the type (arithmetic, geometric, Fibonacci, or custom), and the calculator instantly displays the result you need.
What Can You Do with a Sequence Calculator?
The tool works with numerical patterns of all kinds. After you enter the initial values, it handles three main tasks:
- Compute the n‑th term – get any term in the sequence without writing out all preceding ones. For example, find the 50th term of 5, 9, 13, 17, … in a single step.
- Calculate the sum of the first n terms – add up an arithmetic or geometric progression, even if it’s hundreds of terms long.
- Generate the next terms – supply the first two or three numbers, and the calculator continues the sequence by applying the detected rule.
- Identify the rule – with an auto‑detect option, it checks differences and ratios to tell whether you’re dealing with a linear, exponential, or Fibonacci‑like progression.
No matter if you’re verifying homework, analyzing a series for a project, or just curious about a pattern, the sequence calculator eliminates arithmetic mistakes and saves time.
Types of Sequences You Can Compute
The calculator covers the most frequently used sequences.
Arithmetic sequence – each term increases by a constant difference d. Example: 2, 7, 12, 17, … (d = 5). You need at least two terms to define it.
Geometric sequence – each term is multiplied by a constant ratio r. Example: 3, 6, 12, 24, … (r = 2). For infinite geometric series where |r| < 1, the tool also returns the sum to infinity.
Fibonacci sequence and Lucas variations – a recurrence where each new term equals the sum of the two previous ones. Starting with 1,1 gives the classic Fibonacci series; change the initial pair to obtain Lucas numbers or custom sequences.
Square, cube, and triangular numbers – predefined rules such as n², n³, and n(n+1)/2 are built in. Just set the type and the calculator generates the sequence.
Custom recurrences – you can define your own rule, e.g., a_n = 2*a_{n-1} + 1. Enter the first term(s) and the recurrence relation, and the tool computes as many terms as required.
Formulas the Sequence Calculator Uses
Understanding what happens behind the scenes helps you trust the results.
| Sequence type | n‑th term formula | Sum of first n terms |
|---|---|---|
| Arithmetic | aₙ = a₁ + (n–1)d | Sₙ = n/2 (2a₁ + (n–1)d) |
| Geometric | aₙ = a₁ · rⁿ⁻¹ | Sₙ = a₁ (1 – rⁿ)/(1–r) (r≠1) |
| Fibonacci | aₙ = aₙ₋₁ + aₙ₋₂ | Not a simple closed form; the calculator sums iteratively |
| Triangular | aₙ = n(n+1)/2 | Sₙ = n(n+1)(n+2)/6 |
For an arithmetic progression, the difference d is found directly from any two consecutive terms. For a geometric progression, the ratio r is the quotient of a term and its predecessor. The infinite geometric sum S = a₁/(1–r) is applied automatically when you request an “infinite” calculation and |r| < 1.
Worked Examples
Example 1 – Arithmetic term and sum
Sequence: 5, 9, 13, 17, … (d = 4).
Find the 30th term: a₃₀ = 5 + (29)×4 = 121.
Sum of the first 30 terms: S₃₀ = 30/2 × (5 + 121) = 15 × 126 = 1,890.
Example 2 – Geometric sum to infinity
Sequence: 8, 4, 2, 1, … (r = 0.5).
Infinite sum: S = 8/(1 – 0.5) = 8/0.5 = 16.
Example 3 – Fibonacci-like custom series
Start with 2, 2. The rule: next term = sum of previous two.
Sequence: 2, 2, 4, 6, 10, 16, 26, …
The 10th term, computed step by step, is 178. The calculator does this without iteration on your side.
In each case, entering the first few terms and selecting the appropriate type delivers the result immediately.
Whether you’re preparing for a math exam, checking a formula, or exploring number patterns, the sequence calculator turns tedious manual work into a quick, error‑free process. There’s no download or sign‑up–just open the page and compute.