Geometric Sequence Calculator

Solve aₙ = a₁·rⁿ⁻¹ ; Sₙ = a₁(rⁿ − 1)/(r − 1) step by step — free geometric sequence calculator with worked examples, real-world defaults and instant answers.

aₙ = a₁·rⁿ⁻¹ ; Sₙ = a₁(rⁿ − 1)/(r − 1)

First term a₁Common ratio rTerm number n

n-th term aₙ39,366

Sum of first n terms Sₙ59,048

1
Each term multiplies by r, so the n-th term needs n−1 multiplications:
aₙ = a₁·rⁿ⁻¹ = 2 × 3^9 = 39,366
2
Partial-sum formula (multiply S by r, subtract, telescope):
Sₙ = a₁(rⁿ − 1)/(r − 1) = 2(3^10 − 1)/(3 − 1) = 59,048

Geometric growth is compounding: investments, populations, chain emails. The infinite-sum shortcut a₁/(1−r) only exists when |r| < 1 — the same condition behind 0.999… = 1 and every repeating-decimal-to-fraction conversion.

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Geometric sequence — nth term and partial sum from a₁ and ratio r.

About Geometric Sequence Calculator

The Geometric Sequence Calculator solves aₙ = a₁·rⁿ⁻¹ ; Sₙ = a₁(rⁿ − 1)/(r − 1) for you with the full working shown — every substitution, every unit, every step, exactly the way a good teacher writes it on the board. Geometric growth is compounding: investments, populations, chain emails.

How to use Geometric Sequence Calculator

1Enter your known values in the input fields (sensible real-world defaults are pre-filled).
2The tool substitutes them into aₙ = a₁·rⁿ⁻¹ ; Sₙ = a₁(rⁿ − 1)/(r − 1) and recomputes live.
3Read the answer in the result box, then expand the step-by-step solution to see the full working.
4Copy the method into your notebook, or change inputs to explore how the result behaves.

Why use Geometric Sequence Calculator?

✓Solves aₙ = a₁·rⁿ⁻¹ ; Sₙ = a₁(rⁿ − 1)/(r − 1) instantly as you type — no submit button, no waiting
✓Step-by-step solution shown for every calculation, not just the final answer
✓Realistic example values pre-loaded so you can see a worked example immediately
✓100% free, no sign-up, runs entirely in your browser — your numbers never leave your device
✓Mobile-friendly and fast enough to use mid-homework or mid-lesson

Frequently asked questions

What formula does the Geometric Sequence Calculator use?+

It uses aₙ = a₁·rⁿ⁻¹ ; Sₙ = a₁(rⁿ − 1)/(r − 1). The steps section shows the formula with your actual numbers substituted, so you can follow (and verify) every stage of the calculation rather than trusting a black box.

What should I keep in mind when using this calculator?+

Is the Geometric Sequence Calculator free to use?+

Completely free, with no sign-up or limits. All computation happens client-side in your browser, so it is fast, private, and works even on slow connections once the page has loaded.

Can I use this for homework, teaching or exam prep?+

That is exactly what it is built for. The step-by-step working mirrors how solutions are presented in class, so you can check homework, build worked examples for teaching, or practise method recall before exams.

Embed Geometric Sequence Calculator on your website

Want Geometric Sequence Calculatoron your own site? Paste this snippet into any HTML page — it's free, with no API key or sign-up. The tool loads in an iframe and keeps working exactly as it does here.

Embed code

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