Find the nth term and sum of an arithmetic or geometric sequence, list the terms, and check whether an infinite geometric series converges.
Enter the first term, common difference, and n to see the nth term and sum
Enter the first term, common ratio, and n to see the nth term and sum
This free sequence and series calculator finds the nth term and the sum of the first n terms for both arithmetic and geometric sequences, lists out the individual terms, and checks whether an infinite geometric series converges to a finite sum. An arithmetic sequence adds the same constant (the common difference, d) at every step, while a geometric sequence multiplies by the same constant (the common ratio, r) at every step — two of the most fundamental patterns in mathematics, showing up everywhere from simple interest calculations to compound growth models. Whether you need an arithmetic sequence calculator for a quick homework check or a geometric sequence calculator to explore infinite series convergence, this tool shows the formula and full working behind every result.
The tool offers two modes. Arithmetic Sequence takes a first term (a₁), a common difference (d), and a term number/count (n), then returns the nth term (aₙ) and the sum of the first n terms (Sₙ), along with a listing of the sequence itself. Geometric Sequence takes a first term (a₁), a common ratio (r), and n, returning the nth term, the sum of the first n terms, and — when the series converges (|r| < 1) — the infinite sum (S∞) as well.
This tool is built for students learning sequences and series in algebra or precalculus, teachers preparing worksheets on patterns and progressions, competitive-exam candidates who need instant nth-term and sum calculations, and anyone modeling growth or decay — whether that's compound interest, population change, or a physics problem involving uniformly accelerated motion.
Sequences and series are the mathematical language of patterns that repeat with a constant rule. Arithmetic sequences model steady, linear change — like simple interest or a savings plan with fixed monthly deposits. Geometric sequences model multiplicative, compounding change — like compound interest, population growth rates, or radioactive decay. Understanding when an infinite geometric series converges (versus diverges) is foundational to calculus, engineering, and financial mathematics.
Finance uses arithmetic sequences to model simple interest and loan amortization schedules, and geometric sequences to model compound interest and annuities. Physics describes uniformly accelerated motion with arithmetic sequences and radioactive decay with geometric sequences. Computer science analyzes algorithm growth patterns and recursive data structures using both sequence types. Population modeling projects growth under constant absolute change (arithmetic) or constant percentage change (geometric). Infinite geometric series solve classic problems like Zeno's paradox and fractal perimeter or area calculations.
How this calculator turns a₁, d or r, and n into the nth term and sum
Each term adds the same constant d — a linear pattern that grows (or shrinks) by equal steps.
Each term multiplies by the same constant r — an exponential pattern of growth or decay.
An infinite geometric series only has a finite sum when |r| < 1 — otherwise the terms never shrink toward zero.
From choosing a sequence type to reading the result
Pick Arithmetic Sequence or Geometric Sequence from the mode tabs.
Type the value of a₁, the first term of your sequence.
For arithmetic, enter d (the amount added each step); for geometric, enter r (the factor multiplied each step).
Type the term number or count of terms you want to compute.
The calculator applies the matching nth-term and sum formulas instantly.
Review the nth term, the sum of the first n terms, the infinite sum (geometric only, when convergent), and the listed sequence terms.
A geometric sequence with a₁=2, r=0.5, n=5
A ball drops and bounces to half its previous height each time, starting at 2 meters. Find the height on the 5th bounce, the total distance summed over 5 bounces, and the theoretical infinite total.
Explanation: Even though the sequence continues forever, each term shrinks fast enough (|r|=0.5 < 1) that the running total approaches a finite limit of 4 rather than growing without bound. This is the same mathematical idea behind Zeno's paradox — infinitely many shrinking steps can still add up to a finite total.
What each output actually means
| Output | What It Means | Example |
|---|---|---|
| aₙ (nth term) | The value of the sequence at position n | Arithmetic a₁=1,d=3: a₅ = 13 |
| Sₙ (sum of n terms) | The running total of the first n terms added together | Geometric a₁=2,r=0.5: S₅ = 3.875 |
| S∞ (infinite sum) | The finite limit an infinite geometric series approaches, only when |r| < 1 | a₁=2,r=0.5: S∞ = 4 |
| "Does not converge" (|r| ≥ 1) | No finite infinite sum exists — the series diverges or oscillates forever | a₁=2,r=1.5: S∞ undefined |
Reading a decreasing sequence: a negative common difference (arithmetic) or a ratio between −1 and 1 excluding 0 (geometric) both produce sequences that shrink over time — this is expected behavior, not an error.
Typical ranges: arithmetic sums grow roughly proportional to n² for large n (since terms grow linearly and there are n of them), while convergent geometric sums stay bounded near S∞ regardless of how large n gets.
Manual verification: for an arithmetic sequence, check that consecutive terms differ by exactly d each time; for geometric, check that each term divided by the previous term equals r consistently.
Where nth-term and sum calculations are genuinely useful
Practice arithmetic and geometric sequence problems for precalculus and algebra coursework.
Get instant nth-term and sum results for sequence and series questions under time pressure.
Model compound interest and annuities as geometric sequences, or simple interest as arithmetic.
Project linear or exponential growth trends using arithmetic and geometric sequence patterns.
Describe uniformly accelerated motion (arithmetic) or radioactive decay (geometric) with sequence formulas.
Analyze algorithm growth patterns and recursive structures using sequence and series concepts.
Project population growth under constant absolute (arithmetic) or percentage (geometric) change.
Explore converging infinite series like Zeno's paradox or fractal perimeter and area problems.
Generate instant, step-by-step sequence examples for classroom demonstrations.
Sanity-check a custom sequence-generating function against exact reference values.
Model steady sales growth (arithmetic) or compounding market growth (geometric) scenarios.
Apply arithmetic progressions to structural spacing or geometric progressions to scaling designs.
What this sequence & series calculator does well, and where it has boundaries
Two fundamental sequence patterns, compared side by side
| Property | Arithmetic Sequence | Geometric Sequence |
|---|---|---|
| Rule | Add a constant (d) each step | Multiply by a constant (r) each step |
| nth term | aₙ = a₁ + (n−1)d | aₙ = a₁ × r^(n−1) |
| Sum of n terms | Sₙ = n/2 × (2a₁+(n−1)d) | Sₙ = a₁(1−rⁿ)/(1−r) |
| Growth pattern | Linear (constant absolute change) | Exponential (constant percentage change) |
| Infinite sum possible? | Never (unless d=0) | Yes, if |r| < 1 |
| Real-world example | Simple interest, fixed deposits | Compound interest, population growth |
Summary: This sequence & series calculator gives you an instant, free way to find the nth term and sum of an arithmetic or geometric sequence, list its terms, and check infinite geometric series convergence — all with the formula and steps shown. Pair it with the Factorial Calculator and Prime Number Checker for a fuller toolkit of number theory and algebra calculators.
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