How to use the sequence calculator
Start by choosing Arithmetic or Geometric, then tell the calculator which values you have. It can calculate a term, a finite sum, a common difference or ratio, or the missing entries in a list. Supplied terms are checked against the selected model before a result is shown.
- Choose Arithmetic or Geometric.
- Select First term & common difference/ratio, Two known terms, or List with missing values.
- Enter the displayed values, then optionally fill Optional: also solve for term n and S_n.
- Click Calculate and review Formula & steps. Use Copy result or Download result (.txt) for the generated text.
What each input mode needs
Choose First term & common difference/ratio when you know a_1 and the step between terms. Arithmetic uses Common difference (d); Geometric uses Common ratio (r). The selected mode controls the formula.
Two known terms uses positions and values to derive the missing sequence settings. Enter Position of first known term (m), Value of that term (a_m), Position of second known term (k), and Value of that term (a_k). Positions must be different positive whole numbers.
List with missing values takes comma-separated entries. Mark a gap with _, ?, x, X, or an empty spot between commas. The list needs at least two known terms. The first entry is position 1.
What the result shows
Optional: also solve for term n and S_n accepts a positive whole-number position. The result includes a_n and the sum of the first n terms. Geometric mode also shows an infinite sum when the absolute value of r is less than 1.
Arithmetic: a_1 = 3, d = 4, n = 4
a_4 = 15
S_4 = 36Formula & steps lists the substitutions used for the selected values. A sequence preview shows up to 50 terms. If the requested position is larger, the calculator still calculates that term and shortens the preview. Non-whole values are displayed with at most six decimal places.
Fill missing sequence values
List mode derives one constant difference or ratio from the first two known entries, calculates the first term, and checks every other known entry. If one value disagrees, the error names its position and the value predicted by the selected model. The calculator checks arithmetic and geometric rules only, so it does not infer Fibonacci, squared, recursive, or other rules.
Geometric mode rejects a zero first term and a zero common ratio. A zero known reference term can also prevent the ratio from being derived. Numeric fields accept values such as -2.5; scientific notation and a plus-prefixed value fail the number parser. The two position fields and the optional target cannot exceed 1,000,000,000. Units remain plain numbers, with no unit conversion.
Finite totals and saved results
For a sequence exercise, enter two known terms and inspect the recovered first term, common difference or ratio, and Formula & steps. For a payment or production schedule, List with missing values fills gaps after the matching model is selected.
To get a finite total, enter a target in Optional: also solve for term n and S_n, then read S_n in the result. Copy result places the generated text on the clipboard, while Download result (.txt) creates a text download.
When another math calculator fits better
Use Arithmetic Sequence Calculator when the sequence is arithmetic and you need its term or sum calculation. Choose Geometric Sequence Calculator for geometric terms, finite sums, or the infinite sum. For equations, fractions, matrices, and other topics, browse Math Calculators.
Frequently asked questions
How do I use a sequence calculator with two known terms?
Select Two known terms, enter both positions and values, then click Calculate. The calculator derives the common difference or ratio and works back to the first term.
How can I fill missing sequence values?
Choose List with missing values and separate the entries with commas, using _, ?, x, X, or an empty entry for gaps. At least two known terms must remain, and every other known value must fit the selected model.
Does the geometric sequence calculator show an infinite sum?
It shows an infinite sum when the geometric ratio has an absolute value below 1. It also shows the requested finite sum when you enter a target in Optional: also solve for term n and S_n.
What indexing convention does the sequence calculator use?
The first term is position n = 1, the second is n = 2, and so on. A list's first entry is therefore treated as term 1.