Pick a mode
Four tabs, four questions: list terms and sums, evaluate an infinite series, recover the ratio from two known terms, or find which term hits a target value.
01 — THE TOOL
Four modes. Every geometric answer.
Terms, sums, ratios, and “which term hits this value” — each mode keeps its own numbers, and everything recalculates as you type.
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TRY
Fractions stay exact — try r = 1/3 and watch the working keep perfect fractions.
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The infinite series converges whenever |r| < 1 — try 0.5, 1/3, or −0.25. Anything at |r| ≥ 1 diverges, and the tool will tell you why.
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The positions don’t have to be consecutive — the gap n − m is handled, including the ± case when the gap is even.
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Works with negative and fractional ratios too — signs and parity are checked exactly, not approximately.
02 — HOW TO USE IT
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How to use the calculator
Thirty seconds from question to answer — no manual required.
Enter what you know
First term and ratio for most modes. Decimals, negatives, and fractions like 1/2 or 3/4 all work — the arithmetic stays exact, never approximated.
Read the results
The nth term, the sum, every listed term, a shape chart, and line-by-line working — all recomputed live as you type.
Copy & tweak
One click copies the full worked solution for your notes or homework. Change any input and everything updates instantly.
03 — FAQ
Frequently asked questions
The short answers to the questions students actually ask.
A geometric sequence is a list of numbers where each term after the first comes from multiplying the one before it by a fixed number called the common ratio, r. For example, 2, 6, 18, 54 is geometric with first term a₁ = 2 and r = 3, because 2×3 = 6, 6×3 = 18, and so on. The ratio can also be negative (alternating signs) or a fraction (terms shrinking toward zero).
Use the formula aₙ = a₁ × rⁿ⁻¹ — the first term multiplied by the ratio raised to one less than the position. For a₁ = 2 and r = 3, the 10th term is 2 × 3⁹ = 39,366. The calculator’s Sequence & Sum tab applies this formula with exact fractions and shows every step.
Divide any term by the term before it: r = aₙ₊₁ / aₙ. If the two terms you know aren’t consecutive, use r = (aₙ / aₘ)^(1/(n−m)) — raise the quotient to the power of one over the gap in positions. The Find Ratio tab handles both, including the ± ambiguity when the position gap is even.
For r ≠ 1 the sum is Sₙ = a₁(1 − rⁿ) / (1 − r). When r = 1 every term equals a₁, so the sum is simply n × a₁. Example: 2 + 6 + 18 + … + 39,366 (10 terms, r = 3) gives S₁₀ = 59,048.
Only when the ratio shrinks — that is, |r| < 1. Then the terms approach zero and the infinite sum exists: S∞ = a₁ / (1 − r). If |r| ≥ 1 the terms don’t approach zero, so the series diverges and has no finite sum. Try 8 with r = 1/2: S∞ = 16.
Yes. A negative ratio makes the signs alternate, as in 3, −6, 12, −24 (r = −2). A ratio between −1 and 1 — like 1/2 or −1/3 — makes the terms shrink in magnitude toward zero. The calculator accepts any ratio, including fractions like 3/4, and computes exactly.
No. Every calculation runs locally in your browser — nothing you type is sent to a server, saved, or logged. The tool is free, needs no signup, and works offline once the page has loaded.