Number Sequence Calculator
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What Your Result Means
- Nth Term: The value of the last term in your sequence. For arithmetic sequences it grows linearly; for geometric sequences it grows (or shrinks) exponentially.
- Sum of Series: The total when you add up all terms from the first through the nth. Useful in finance (annuity payments), physics (distance under constant acceleration), and many textbook problems.
- Sequence (first 10): A preview of the actual terms so you can visually confirm the pattern matches your expectations before relying on the nth term or sum.
How This Calculator Works
You enter a first term, choose arithmetic or geometric, provide the common difference (d) or common ratio (r), and specify how many terms. For arithmetic sequences, the nth term is a₁ + (n−1)d and the sum is (n/2)(2a₁ + (n−1)d). For geometric sequences, the nth term is a₁ × r^(n−1) and the sum is a₁(1 − r^n)/(1 − r). It assumes exact values with no rounding until display.
Quick Questions
What is the difference between arithmetic and geometric?
An arithmetic sequence adds a constant amount each step (2, 5, 8, 11 — adding 3). A geometric sequence multiplies by a constant each step (3, 6, 12, 24 — multiplying by 2). Arithmetic grows linearly, geometric grows exponentially.
What happens if the common ratio is 1?
Every term equals the first term, and the sum is simply the first term times the number of terms. The calculator handles this as a special case to avoid dividing by zero in the geometric sum formula.
Can I use negative differences or ratios?
Yes. A negative common difference creates a decreasing arithmetic sequence. A negative ratio creates an alternating-sign geometric sequence (e.g., 1, −2, 4, −8). Both are mathematically valid and the formulas still apply.
Where are sequences used in real life?
Arithmetic sequences model equal payments, uniform motion, and evenly spaced measurements. Geometric sequences model compound interest, population growth, radioactive decay, and any process where a quantity changes by a fixed percentage each period.
Sources
- Wikipedia — Arithmetic Progression (formulas, derivation, properties)
- Wikipedia — Geometric Series (finite and infinite sum formulas)
- Khan Academy — Sequences (interactive lessons and practice problems)
Method & review
Estimate only. Results reflect your inputs and standard formulas. Double-check important decisions independently.