Sequence & Series Calculator — Arithmetic, Geometric & Fibonacci
Step-by-Step Solution
By Amanda Reeds, Content Researcher ·
A sequence is an ordered list of numbers that follows a fixed rule, and a series is the sum you get when you add those numbers together. This sequence and series calculator finds the nth term and sum of arithmetic and geometric sequences, generates Fibonacci sequences, and runs convergence tests, with every step shown.
It's built for algebra and calculus students, tutors, and anyone who needs to check summation work without reaching for a symbolic math app. Whether you're finding the common difference in an arithmetic sequence, checking whether a geometric series converges before you try to sum it to infinity, or just want to see the first fifteen Fibonacci numbers without counting on your fingers, the five modes below cover the most common sequence and series problems in one tool. Type your numbers, press Solve, and the calculator walks through the same formula you'd use by hand.
Five modes live in one widget: Arithmetic, Geometric, Fibonacci, Convergence Tests, and a general Series Solver that accepts almost any term formula written in n, so you're not hunting across five different calculator pages for one homework set.
What You Can Do With This Sequence & Series Calculator
Arithmetic Sequence & Series Calculator
Arithmetic mode applies the arithmetic sequences formula a_n = a1 + (n-1)d to find any term, and the arithmetic series formula to sum the first n terms. Switch the "Find" option to solve for the common difference or the number of terms instead.
- Find the nth term and the sum in one pass
- Find the common difference (d) from two known terms
- Find how many terms (n) reach a target value
Geometric Sequence & Series Calculator
Geometric mode uses the geometric sequence formula a_n = a1 · r^(n-1) and the finite geometric series formula, then can flip to solve for the common ratio when you already know two terms in the sequence.
- Finite nth term and sum, with full working
- Find the common ratio (r) from a1 and a later term
- Same core logic behind our Quadratic Formula Calculator's step engine
Infinite Geometric Series & Convergence
The Infinite Sum option in Geometric mode applies S = a1/(1-r), and only returns a value when the common ratio is between -1 and 1. Outside that range, the calculator tells you the series diverges instead of forcing a number out of a formula that doesn't apply.
- Instant infinite geometric series calculator with steps
- Clear convergent-vs-divergent verdict, not just a number
- Works alongside the Convergence Tests mode below
Fibonacci Series Calculator
Fibonacci mode generates the sequence up to any number of terms using the formula for Fibonacci sequence, F(n) = F(n-1) + F(n-2), and shows the running list alongside the specific term you asked for. According to Britannica, Leonardo of Pisa introduced the sequence into Western mathematics in 1202, in his book Liber Abaci, through a problem about the growth of a rabbit population.
- Generate the first N Fibonacci numbers instantly
- Highlights the exact term you're checking
- Notes Binet's closed-form shortcut in the steps
Series Convergence Tests
Convergence Tests mode runs the geometric series test, the p-series test, and the alternating series test (Leibniz test) on the series type you pick, each with the specific condition that decides convergent and divergent behavior.
- Sequence convergence calculator for geometric and p-series forms
- Alternating series test calculator with steps
- Explains when the ratio, root, and integral tests apply instead
General Series Solver
Series Solver mode accepts almost any term formula, like 1/n^2 or 2^n/n!, lists the individual terms, adds up the partial sum, and can estimate an infinite sum numerically when the terms shrink toward zero.
- Works as a general summation series formula evaluator
- Handles polynomial, fraction, and factorial-style terms
- Flags series whose terms don't shrink, since those can't converge
The Formulas Behind The Calculator
Every result this calculator produces comes from one of the standard formulas below. According to Wikipedia's entry on arithmetic progressions, an arithmetic sequence is any list of numbers where each term after the first is formed by adding a fixed value to the one before it, a pattern that dates back to work by Carl Friedrich Gauss on summing consecutive integers. The convergence rules below follow the standard treatment in Wikipedia's entry on convergent series, including the geometric and p-series tests used most often in an introductory calculus course.
| Formula | Rule | Used For |
|---|---|---|
| Arithmetic nth Term | a_n = a1 + (n-1)d | Any term in an arithmetic sequence |
| Arithmetic Series Sum | S_n = n/2 · (2a1 + (n-1)d) | Sum of the first n arithmetic terms |
| Geometric nth Term | a_n = a1 · r^(n-1) | Any term in a geometric sequence |
| Geometric Series Sum | S_n = a1(1-r^n)/(1-r) | Sum of the first n geometric terms, r ≠ 1 |
| Infinite Geometric Sum | S = a1/(1-r) | Only valid when |r| < 1 |
| Fibonacci Recurrence | F(n) = F(n-1) + F(n-2) | Every Fibonacci number after F(0), F(1) |
| p-Series Test | Σ 1/n^p converges only if p > 1 | Testing series that look like 1/n^p |
How To Use The Sequence & Series Calculator
Pick the mode that matches your problem first. Arithmetic and Geometric cover the two most common sequence types taught in algebra, Fibonacci handles the recursive number sequence on its own, Convergence Tests checks whether an infinite series settles on a value, and Series Solver takes almost any term formula you can write in one line.
- Choose a mode from the row above the screen.
- Type the values the mode asks for. For example, an arithmetic sequence starting at 5 with a common difference of 3 over 12 terms enters as a1 = 5, d = 3, n = 12, which returns a_12 = 38 and a sum of 258.
- Use the "Find" dropdown to switch what the calculator solves for, such as finding d instead of the sum.
- Press Solve to see the full step-by-step breakdown below the widget.
- In Series Solver mode, type a term formula like 1/n^2 or 2^n/n!, using n as the variable, then choose whether to see the partial sum or an infinite-sum estimate.
Quick Action Steps
- Pick Arithmetic, Geometric, Fibonacci, Convergence Tests, or Series Solver.
- Enter the known values for that mode's formula.
- Press Solve and read the term-by-term working, not just the final number.
⚠ Watch Out For This
An infinite geometric sum only exists when the common ratio r is strictly between -1 and 1. Plugging r = 1.5 or r = -2 into S = a1/(1-r) still spits out a number, but that number is meaningless because the series diverges. The calculator checks this condition before returning a result instead of returning a false answer.
Why AceCalculator Is Different And Better
Most sequence and series calculators online give you a final number and stop there. AceCalculator focuses on showing the working, matching classroom notation, and staying fast on any device.
- ✓ Full step-by-step solutions in every mode
- ✓ Five modes in one tool: arithmetic, geometric, Fibonacci, convergence, general solver
- ✓ Built-in partial-sum chart for series and convergence checks
- ✓ Clean, classroom-style sequence and series notation
- ✓ Mobile-friendly and responsive on any device
- ✓ No downloads, no accounts, no signup required
- ✓ Runs entirely in your browser, nothing is stored or sent to a server
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Content Researcher · AceCalculator
Amanda Reeds researches and builds AceCalculator's math tools, with a focus on matching each calculator's step-by-step output to standard classroom notation. She has written and verified the working behind AceCalculator's algebra, quadratic, and equation-solving calculators.
Frequently Asked Questions
Yes. Arithmetic and geometric sequences and series, Fibonacci generation, convergence testing, and the general series solver are all free, with no account, signup, or download required.
The nth term of an arithmetic sequence is a_n = a1 + (n-1)d, where a1 is the first term and d is the common difference. The sum of the first n terms is S_n = n/2 · (2a1 + (n-1)d).
The nth term of a geometric sequence is a_n = a1 · r^(n-1), where a1 is the first term and r is the common ratio. The sum of the first n terms is S_n = a1(1-r^n)/(1-r) for r ≠ 1.
Each Fibonacci number is the sum of the two before it: F(n) = F(n-1) + F(n-2), starting from F(0)=0 and F(1)=1. Binet's closed-form formula, using the golden ratio, gives the same value without listing every earlier term.
It depends on the series type. A geometric series Σar^n converges only when |r| < 1. A p-series Σ1/n^p converges only when p > 1. For other series, the ratio, root, integral, or alternating series tests each check a different behavior of the terms as n grows large.
A sequence is an ordered list of numbers, such as 2, 4, 6, 8. A series is what you get when you add those terms together, such as 2+4+6+8. Every series is built from a sequence, but a sequence on its own is never summed.
The alternating series test, also called the Leibniz test, states that a series with terms that alternate in sign converges if the absolute value of each term decreases and the terms approach zero. It only proves convergence, never divergence.
Yes, for an infinite geometric series with |r| < 1, using S = a1/(1-r). For other series entered in the general Series Solver mode, the calculator estimates the value numerically from a large partial sum and flags whether the terms are shrinking toward zero.
Important Disclaimer
AceCalculator's sequence and series calculator is provided for informational and educational purposes only. While every step follows standard sequence and series formulas, results are not a substitute for verification by a qualified instructor where accuracy is critical, especially for numerical estimates of infinite sums. See our full Disclaimer and Terms and Conditions.