Geometric Progression Generator

A geometric progression multiplies instead of adding: every term is the previous term times a fixed value, called the common ratio. 2, 6, 18, 54 is a geometric progression with first term 2 and ratio 3. This is the mathematics of compound interest, population growth, viral spread and half-life — anything that grows or decays by a percentage per step.

Geometric Progression Generator creates the whole sequence instantly. Give it a first term, a common ratio and a length, and it applies the nth-term formula aₙ = a₁ × r^(n−1) to every position. Ratios above 1 make the sequence explode upwards, ratios between 0 and 1 make it shrink, and negative ratios make the signs alternate. When both inputs are whole numbers the tool uses exact BigInt arithmetic, so even 500 terms of a fast-growing sequence stay precise to the last digit — and the sum formula Sₙ = a₁ × (1 − rⁿ)/(1 − r) is verified for you in the output.

• First term, ratio, count: Three inputs — a₁, r and n — cover every geometric progression.
• Exact BigInt arithmetic: Whole-number inputs produce arbitrary-precision results, so huge terms are never rounded.
• Fractional and negative ratios: Ratios like 0.5 or −2 are handled correctly, with clean decimal formatting.
• Running total option: Optionally watch the accumulated sum grow, and always receive the final sum Sₙ of the sequence.
• Live output: The sequence regenerates as you type, making it easy to compare ratios side by side.
• Copy or download: One click copies the sequence, or save it as a plain text file.
• Term-by-term labelling: Every term is marked with its position (a₁, a₂, a₃…), ideal for homework and verification.
• Free and private: Runs entirely in your browser — nothing is uploaded, stored or logged.

1. Enter the first term (a₁). The starting value of the sequence — for example 2.
2. Enter the common ratio (r). The multiplier applied to each term. Use 3 for tripling, 0.5 for halving, or −2 for an alternating sequence.
3. Choose how many terms (n). The generator produces exactly this many values, from a₁ to aₙ.
4. Toggle the running sum if needed. Tick "Show the running sum" to see the total accumulate term by term.
5. Read the output. Terms appear instantly, labelled by position, with the final sum Sₙ on the last line.
6. Copy or download. Copy to the clipboard or download the sequence as a text file.

Example 1 — Doubling experiment. Start with a₁ = 1 and ratio r = 2, and generate 12 terms: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048. That is also the classic story of the grains of rice on a chessboard, where the 64th term alone is astronomical.

Example 2 — Compound interest. A 5% monthly return multiplies a balance by 1.05 each month. Set a₁ = 1000 and r = 1.05 for 24 terms to see the money grow from 1,000 to over 3,225 — the output makes the exponential curve visible.

Example 3 — Half-life. A radioactive sample decays by 50% per day: a₁ = 100, r = 0.5. The generator shows 100, 50, 25, 12.5, 6.25… demonstrating why half-life decay never quite reaches zero.

Example 4 — Alternating signs. With r = −3, the sequence 1, −3, 9, −27, 81… alternates sign. Useful for understanding alternating series and interference patterns in engineering.

• Instant exponential sequences: No manual multiplication chains — every term follows from the nth-term formula automatically.
• Exact for huge values: BigInt precision keeps fast-growing sequences exact, which floating-point calculators lose quickly.
• Sum built in: The finite sum Sₙ is computed and displayed, with the geometric sum formula applied for you.
• Explores real-world growth: Compound interest, half-life, doubling and decay are all one ratio away — try 1.05, 0.5 and 2 side by side.
• Great for learning: Labelled terms and an optional running total make the growth pattern obvious at a glance.
• Free and private: Runs entirely in the browser; nothing is uploaded, stored or logged.