Number Sequence Calculator
Identify and continue arithmetic and geometric sequences. Find the nth term and sum of the first n terms.
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How to use this calculator
Arithmetic sequences add a constant difference d; geometric sequences multiply by a constant ratio r.
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Enter the first and second terms of your sequence.
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Enter n to find the nth term of the sequence.
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Enter N to find the sum of the first N terms.
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Both arithmetic (constant difference) and geometric (constant ratio) results are shown.
Frequently asked questions
How do I know if a sequence is arithmetic or geometric?
Check if consecutive terms have a constant difference (arithmetic) or constant ratio (geometric). 3, 7, 11, 15 → difference = 4 each time → arithmetic. 3, 6, 12, 24 → ratio = 2 each time → geometric. Some sequences are neither.
What is the sum of an infinite geometric series?
If |r| < 1, the infinite geometric series converges: S = a₁ / (1 − r). For example, 1 + 0.5 + 0.25 + … = 1 / (1 − 0.5) = 2. If |r| ≥ 1, the series diverges (sum is infinite).
What is the Fibonacci sequence?
The Fibonacci sequence starts 1, 1, 2, 3, 5, 8, 13… where each term is the sum of the two preceding ones. It is neither arithmetic nor geometric. It appears in nature (spiral shells, flower petals, leaf arrangements) and has a connection to the golden ratio φ.
What is an arithmetic series vs arithmetic sequence?
A sequence is an ordered list of terms. A series is the sum of terms. The sequence 3, 7, 11, 15 has a related series 3 + 7 + 11 + 15 = 36. Gauss's formula for the sum of the first n integers: S = n(n+1)/2 is a special case.
Arithmetic and geometric sequences explained
How to use the number sequence
This number sequence gives you instant, accurate results — no registration or download required. Enter your values above and get your result in seconds. The tool is free, works on all devices, and keeps your data private — nothing is stored or shared.
How the number sequence works
The number sequence uses standard formulas used in financial planning, budgeting, and investment decisions. Enter your inputs, and the tool calculates the result instantly in your browser. No server-side processing means your data stays on your device. Results update in real time as you change inputs.
Arithmetic sequences in everyday life
Bus schedules (every 12 minutes), stadium seating (each row has 2 more seats than the last), and salary increments (annual raises of a fixed amount) are all arithmetic sequences. Gauss famously summed 1 to 100 as an arithmetic series: S = 100 × 101 / 2 = 5,050.
Geometric sequences in finance and science
Compound interest, population growth, radioactive decay, and Moore's Law all follow geometric sequences. If you invest $1,000 at 8% annual return, the balance after n years is 1000 × 1.08^n — a geometric sequence with ratio 1.08.
Convergence of infinite geometric series
Zeno's paradox (Achilles and the tortoise) is resolved by an infinite geometric series with |r| < 1 converging to a finite sum. The series 1 + 1/2 + 1/4 + 1/8 + … = 2 exactly. This convergence underpins integral calculus and is the basis of how loans, annuities, and perpetuities are valued.
Number sequence: how it works
Built on well-established mathematical principles, this tool delivers accurate results for students, researchers, and professionals. Enter your values and get instant clarity without specialist software or manual arithmetic.
Who uses this tool?
Teachers, students, engineers, and data analysts use it to verify calculations, check homework, and solve problems faster. It is intuitive enough for beginners while comprehensive enough for professional use.
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