Sequence Calculator
Find the nth term, sum, and infinite series — instantly, with steps.
What are you solving?
Inputs
Formula, filled in
Step by step
About the Sequence Calculator
The Sequence Calculator is a professional mathematical tool designed to help you quickly calculate various properties of arithmetic and geometric sequences. Whether you're working on high school math homework, college calculus courses, or financial investment applications, this calculator provides accurate results and detailed calculation steps. Our formulas are verified by authoritative sources including Wolfram MathWorld and Khan Academy, ensuring 100% accuracy.
How to use it
- Pick a sequence type: arithmetic, geometric, or compare both
- Enter the first term a₁, plus d (arithmetic) or r (geometric)
- Choose how many terms (n)
- Read the nth term, the sum, and — if geometric — the infinite series total
- Check the chart, the step-by-step breakdown, and the full term list
Real-world examples
🏦 Equal-principal loan (arithmetic)
Borrow $600,000, repay over 20 installments, 1% monthly interest — each installment drops by a fixed amount. First month interest = 600,000 × 0.01 = $6,000. Period n interest = 6,000 - (n-1) × 300. Total interest = S₂₀ = 20(6,000+300)/2 = $63,000.
💰 Compound savings (geometric)
Deposit $100,000 at 2% annual compound interest — balance grows by a fixed ratio each year. After 5 years, total = 100,000 × 1.02⁵ ≈ $110,408.
🏙️ Population growth (geometric)
A city of 1,000,000 growing 3% per year — same ratio-based growth model. After 10 years, population ≈ 1,344,000.
Formula reference
Arithmetic: aₙ = a₁ + (n−1)d, sum Sₙ = n(a₁+aₙ)/2 = n[2a₁+(n-1)d]/2.
Geometric: aₙ = a₁ × r^(n−1), sum Sₙ = a₁(1-r^n)/(1-r) (r≠1).
Infinite geometric series: S = a₁ / (1−r), only when |r| < 1.
FAQ
What's the difference between arithmetic and geometric?
Arithmetic sequences add a fixed amount each step. Geometric sequences multiply by a fixed ratio each step.
When does an infinite geometric series converge?
Only when the absolute value of the common ratio r is less than 1. Otherwise the sum grows without bound.
Sequences in everyday life
Sequences are not just textbook knowledge—they're tools used everywhere in life: loan calculations, compound interest, population forecasting, radioactive decay, and more.
