Sequence Calculator

How to use the sequence calculator

What this calculator does

This calculator generates a five-term window from one of ten standard numerical sequences. It evaluates the selected rule at the chosen Initial position (n), then continues through the next four integer indexes. The result is useful for checking homework, exploring growth patterns, preparing a small table of values, or verifying a formula before graphing it. It does not infer an unknown rule from arbitrary terms, prove a theorem, or sum the sequence as a series. OpenStax's explanation of sequences, explicit formulas, and indexing provides a useful conceptual foundation.

When to use it

• Check five consecutive terms of an arithmetic or geometric sequence from a specified index.
• Compare how polynomial, exponential, recursive, prime, factorial, and figurate sequences grow.
• Create a clean term table and discrete plot for a lesson, worksheet, or calculation note.
• Export inputs and typed results to an Excel workbook for further analysis.

How to calculate

1. The calculator opens with a complete arithmetic demonstration: a₀ = 3, d = 4, and n = 5. Its first-open results and Excel workbook are ready immediately.
2. Select a Sequence type. Arithmetic and geometric choices reveal two additional required controls; the other standard sequences need only an index.
3. Enter the Initial position (n) as a whole number in the permitted range. Replace the sample Initial term (a₀) and Common difference (d) or Common ratio (r) when those fields are active.
4. Read the first displayed term, final displayed term, net change, minimum, maximum, explicit rule, five-term cards, plot, and detail table. All update live.
5. Select Download Excel to export the current validated model. Select Reset to clear the demonstration and calculated content; Excel export is then disabled until a complete valid state is entered again.

Input guide

Sequence type is required and accepts one named rule: Arithmetic, Geometric, Square numbers, Cube numbers, Triangular numbers, Prime numbers, Fibonacci numbers, Powers of 2, Factorials, or Star numbers. Arithmetic uses a constant added difference; geometric uses a constant multiplier. The remaining choices use fixed standard definitions. A common mistake is expecting a geometric sequence to add the ratio rather than multiply by it.

Initial position (n) is required and accepts a nonnegative whole number written with optional U.S. thousands grouping, such as 1,000. Star numbers begin at n = 1 in this calculator. Factorials must keep the last displayed index at or below 170, powers of 2 at or below 1023, and prime indexes at or below 10,000. Fractions, decimal commas such as 1,5, and scientific notation are rejected. A higher index moves the five-term window later in the sequence and can increase values dramatically.

Initial term (a₀) is required only for arithmetic and geometric sequences. It accepts a signed U.S.-style decimal such as 3, -2.5, or 1,250.75. It sets the term at index zero. Changing it shifts every arithmetic term by the same amount and scales every geometric term. Do not enter a currency symbol, percent sign, or decimal comma.

Common difference (d) is required for arithmetic sequences. It is a signed decimal added at each step; for example, d = 4 produces a steady increase of four, while d = -4 produces a steady decrease. Common ratio (r) occupies the same control for geometric sequences and multiplies each term; for example, r = 2 doubles values, r = 0.5 halves them, and a negative ratio alternates signs. Zero is allowed, but geometric values after a positive index then become zero.

Output guide

Term aₙ is the exact first value in the current five-term window. Final displayed term is the value four indexes later. Net change is final minus first, so a negative number indicates an overall decrease and zero means the two endpoints match. Minimum term and Maximum term identify the range within the five displayed values, not across the infinite sequence. The formula states the rule and current parameters.

The Five-term window, Term progression, and Term details all represent the same five canonical values. The plot treats indexes as discrete points. The table's Position column names each a-index, Term value gives its number, and Change from previous gives the difference between adjacent displayed rows. The summary pills report type, start index, term count, and trend. These are exact identities for the selected formula, subject only to finite floating-point representation for non-integer decimals.

Worked example

With the startup arithmetic values a₀ = 3, d = 4, and n = 5, the explicit formula is aₙ = a₀ + n·d. Therefore a₅ = 3 + 5×4 = 23. Continuing by adding four gives 27, 31, 35, and 39 at indexes 6 through 9. The final displayed term is 39, the net change is 39 – 23 = 16, the minimum is 23, and the maximum is 39. These same values appear in the first-open chart, table, and workbook.