Sequence & Series Calculator
Compute the nth term and the sum of an arithmetic or geometric sequence. Toggle the type, fill in the first term, the common difference (arithmetic) or ratio (geometric), and how many terms — see every step and the sum to infinity when it converges.
Sequence vs series — what's the difference?
A sequence is an ordered list of numbers generated by a rule, like 2, 5, 8, 11, 14, …. Each number is called a term, and you index them a₁, a₂, a₃, …
A series is the running total of a sequence: a₁ + a₂ + a₃ + …. The sum of the first n terms is written Sₙ. So the sequence describes the pattern; the series describes the accumulation.
The sequences this tool handles, piece by piece
Each card explains one kind of sequence the tool builds — how each term is generated, the closed form for the n-th term and the partial sum, and (for geometric) the special case when the sum converges.
Arithmetic — add the same amount each step
Each term is the previous term plus a fixed common difference d. The n-th term and the sum of the first n terms both have clean closed formulas.
Geometric — multiply by the same ratio each step
Each term is the previous term times a fixed common ratio r. Terms grow (|r|>1), shrink (|r|<1) or alternate (r<0).
Sum to infinity — only for shrinking geometrics
When |r| < 1 the terms shrink to zero fast enough that the partial sums converge to a finite limit. Otherwise the series diverges.
Features of this calculator
- Arithmetic and geometric modes with the correct formula applied automatically
- Reports the nth term aₙ and the sum of the first n terms Sₙ side by side
- Optional sum to infinity for geometric series — with an explicit check that |r| < 1
- Renders the first few terms as a small bar chart so patterns and sign changes are visible
- Handles negative and fractional inputs — negative ratios produce alternating signs
- Full step-by-step working showing the substitution into the standard formulas
Frequently asked questions
+What is a₀ vs a₁ — do sequences start at 0 or 1?
Both conventions exist. This calculator uses a₁ as the first term (so n = 1 gives you a₁ itself). If a textbook starts at a₀, just shift the index: their a₀ is our a₁, and their formula (n·d instead of (n−1)·d) is an equivalent restatement.
+Why does the geometric sum blow up when r = 1?
The formula Sₙ = a₁·(1 − rⁿ)/(1 − r) has a zero in the denominator when r = 1. But r = 1 means every term equals a₁, so the sum is simply n·a₁ — the calculator applies that special case directly.
+Can I recover the common difference from two terms?
Yes. For an arithmetic sequence, d = (aₙ − aₘ) / (n − m). For a geometric one, r = (aₙ / aₘ)^(1 / (n − m)). Plug the recovered d or r back into this calculator to explore the rest.
+Is 0.999… really equal to 1?
Yes — and geometric series prove it. 0.999… = 9/10 + 9/100 + 9/1000 + … is geometric with a₁ = 9/10 and r = 1/10, so S∞ = (9/10) / (1 − 1/10) = (9/10) / (9/10) = 1.