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Algebra

Geometric Sequence

nth term and sum. Free online Geometric Sequence. Calculate geometric sequence online — fast, accurate, mobile-friendly, no signup needed.

aₙ
486
Sₙ
728

Derivation

  1. ├── 01Givena1 = 2, r = 3, n = 6
  2. ├── 02Formulaaₙ: t × (a)^(n-1)
  3. ├── 03Substitutet × (a)^(6-1)
  4. ├── 04Compute aₙ486
  5. ├── 05FormulaSₙ: t × ((a)^(n)-1) / (a-1)
  6. ├── 06Substitutet × ((a)^(6)-1) / (a-1)
  7. └── 07Compute Sₙ728
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§01What is

Understanding the Geometric Sequence

The Geometric Sequence computes aₙ from 3 inputs: first term, ratio, term n. nth term and sum.

Algebra is the art of solving for the unknown. Rearranging a formula to isolate the variable you actually need is the single most common real-world math skill — and doing it with real numbers under time pressure is where errors happen. The Geometric Sequence sits in that toolkit — it nth term and sum. Enter your numbers above and the result updates instantly; every step of the math is shown in the Derivation panel so you can see exactly how the answer was reached.

§02The Formula

How it’s calculated

aₙ = t × (a)^(n-1) | Sₙ = t × ((a)^(n)-1) / (a-1)

Where

a1
First term
r
Ratio
n
Term n
aₙ
Output value
Sₙ
Output value
§03Practical Example

Step-by-step walkthrough

Scenario

Apply the formula to a realistic set of inputs: First term = 2, Ratio = 3, Term n = 6.

  1. 01Start by noting the input — First term: 2.
  2. 02Start by noting the input — Ratio: 3.
  3. 03Start by noting the input — Term n: 6.
  4. 04Substitute these values into the formula: aₙ = t × (a)^(n-1) | Sₙ = t × ((a)^(n)-1) / (a-1)
  5. 05Compute aₙ: the calculator returns 486.
  6. 06Compute Sₙ: the calculator returns 728.
  7. 07Cross-check the answer by opening the Derivation panel above — every line of math is shown so you can follow the computation end-to-end.
§04Variants

Common Geometric Sequence Problems

The formula gets rearranged depending on which variable you need. Here are the patterns you’ll run into in the real world — find the one that matches your problem and follow the worked steps.

01 · PATTERN

First term halved

a1 = 1 (from 2)

Keep every other input at its default and halve the first term. See how aₙ responds.

  1. 01New First term: 1
  2. 02Baseline aₙ: 486
  3. 03New aₙ: 243
  4. 04aₙ decreases by 50% → use this sensitivity to plan for real-world variation.
02 · PATTERN

First term doubled

a1 = 4 (from 2)

Keep every other input at its default and double the first term. See how aₙ responds.

  1. 01New First term: 4
  2. 02Baseline aₙ: 486
  3. 03New aₙ: 972
  4. 04aₙ increases by 100% → use this sensitivity to plan for real-world variation.
03 · PATTERN

Ratio halved

r = 1.5 (from 3)

Keep every other input at its default and halve the ratio. See how aₙ responds.

  1. 01New Ratio: 1.5
  2. 02Baseline aₙ: 486
  3. 03New aₙ: 15.1875
  4. 04aₙ decreases by 96.9% → use this sensitivity to plan for real-world variation.
04 · PATTERN

Ratio doubled

r = 6 (from 3)

Keep every other input at its default and double the ratio. See how aₙ responds.

  1. 01New Ratio: 6
  2. 02Baseline aₙ: 486
  3. 03New aₙ: 15552
  4. 04aₙ increases by 3100% → use this sensitivity to plan for real-world variation.
§05FAQ

Frequently asked questions

Yes. The calculator implements the standard formula as documented and returns exact floating-point results. No approximations are used unless noted in the formula.
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