Sequences & series

Sequence Calculator

Find a term and a finite sum from a starting value and a constant difference or ratio.

Nth term

Enter your sequence rule and number of terms, then select Calculate.

≈ marks rounding. Calculations run in your browser. Press Enter to calculate. Successful results appear in this tab's Recent calculations.

Sequence and sum examples

Arithmetic

Add 3, starting at 2

aₙ = a₁ + (n − 1)d

The terms begin 2, 5, 8, 11. Term 10 is 2 + 9 × 3 = 29. The first ten terms add to 10 × (2 + 29) ÷ 2 = 155.

Geometric

Five terms, starting at 3 and doubling

aₙ = a₁ × rⁿ⁻¹

With a₁ = 3, r = 2, and n = 5, the terms are 3, 6, 12, 24, 48. Term 5 is 48, reached after four multiplications. The finite sum is 93.

Infinite geometric series

Halve 8 again and again

S∞ = a₁ ÷ (1 − r)

With a₁ = 8 and r = 0.5, the first five terms sum to 15.5. Continuing indefinitely gives a convergent infinite sum of 16. The finite and infinite sums answer different questions.

Formulas, indexing, and calculation limits

For an arithmetic sequence, aₙ = a₁ + (n − 1)d, and the finite sum is Sₙ = n(a₁ + aₙ)/2. A common difference of zero gives a constant sequence. Negative differences decrease each term by a fixed amount.

For a geometric sequence, the next term is the preceding term multiplied by r. For n ≥ 2, aₙ = a₁ × rⁿ⁻¹. When r ≠ 1, Sₙ = a₁(1 − rⁿ)/(1 − r). When r = 1, every term is a₁ and Sₙ = n × a₁. The first term is always the entered a₁; with r = 0, later terms are zero. This avoids treating the first term as a separate 0⁰ calculation.

A geometric series with a nonzero first term has a finite infinite sum only when |r| < 1. Negative ratios inside that interval also converge. At r = −1, a nonzero sequence alternates without a convergent sum. If a₁ = 0, the recurrence gives only zero terms for every ratio, so its infinite sum is zero.

Enter 1–1000 terms. The first term and difference or ratio each accept at most 64 characters, 15 integer digits, and 10 decimal places. Use decimal values rather than fractions, expressions, or scientific notation. The term count accepts digits only, with surrounding whitespace allowed.

The calculator uses exact fractions throughout the recurrence. To keep the work and answers bounded, every generated term and partial sum must have no more than 200 digits in each reduced numerator and denominator. Inputs exceeding that result limit receive an error, even if a later cancellation could make the final sum smaller. Smaller ratio magnitudes can still create large denominators; fewer terms may be needed.

Only the decimal display is rounded. Halfway values round away from zero, trailing zeros are removed, and ≈ marks an inexact display. A very small nonzero value can appear as ≈ 0. Expand the exact answers to see the full fraction. Changing display precision does not create a new recent-calculation entry. Only the preview is limited to ten terms; the nth term and finite sum use the entire requested count.

This tool applies the rule you supply. It does not infer an unknown sequence from a list of numbers, solve for a missing difference or ratio, or evaluate general series.

References: OpenStax, Algebra and Trigonometry 2e — arithmetic sequences, geometric sequences, and series and their notations.

Common questions

What is the difference between a sequence and a series?

A sequence is an ordered list of terms. A series adds those terms. For the sequence 2, 5, 8, the third term is 8, while the sum of the first three terms is 15.

Can the difference or ratio be negative?

Yes. A difference of −3 subtracts 3 each time. A ratio of −2 multiplies each term by −2, changing the sign of every nonzero term.

Why does a ratio of zero still give a nonzero sum?

The starting term is included. If a₁ = 7 and r = 0, the terms are 7, 0, 0, and so on. Every finite sum and the infinite sum equal 7.

Does a finite sum require a ratio less than one?

No. A finite geometric sum can use any real ratio within the calculator's input and result limits. The condition |r| < 1 concerns the infinite series when the first term is nonzero.