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Geometric sequence — The sum of the first n terms (Sₙ)

Find the sum Sₙ of the first n terms.

Geometric sequence — The sum of the first n terms (Sₙ)

Geometric sequence · Each term is multiplied by the same number.

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Find the sum Sₙ of the first n terms.

Formula used for this answer:

Sₙ = a₁(1 − rⁿ) ÷ (1 − r), when r ≠ 1Sₙ = na₁, when r = 1
  • a₁ = first term
  • r = common ratio (multiplier each time)
  • n = term number
  • Sₙ = sum of the first n terms

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Result

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Find the sum Sₙ of the first n terms.

  1. First term (a₁): Any finite number.
  2. Common ratio (r): The multiplier; it may be zero, one, or negative.
  3. Term number (n): A positive whole number.
  4. Calculate: Result — The sum of the first n terms (Sₙ).
  5. Copy result · Clear

Formula used for this answer:

Sₙ = a₁(1 − rⁿ) ÷ (1 − r), when r ≠ 1

Sₙ = na₁, when r = 1

  • a₁ = first term
  • r = common ratio (multiplier each time)
  • n = term number
  • Sₙ = sum of the first n terms

Values substituted

First term (a₁) = 2 · Common ratio (r) = 3 · Term number (n) = 4 → The sum of the first n terms (Sₙ) = 80

First term (a₁) = 5 · Common ratio (r) = 1 · Term number (n) = 6 → The sum of the first n terms (Sₙ) = 30

Precision, signs, and limits

  • n must be a positive whole number.
  • Use n no greater than 1,000,000,000.
  • The display removes unnecessary trailing zeros and uses scientific notation for very large or very small nonzero results. Calculation retains more precision than the display.
  • A geometric ratio extremely close to 1 uses a numerically stable finite-sum calculation. Inputs that overflow JavaScript numeric range are rejected.
  • Only the first six terms are previewed. The tool does not infer missing patterns, parse formulas, draw graphs, save history, or calculate infinite series.
  • These inputs produce a result outside the supported numeric range. Use smaller values.

Frequently asked questions

First term (a₁)?

Any finite number.

Common ratio (r)?

The multiplier; it may be zero, one, or negative.

Term number (n)?

A positive whole number.

Why is a geometric ratio of 1 handled separately?

The usual sum formula would divide by 1 − r = 0. When r = 1, every term equals the first term, so the sum is n times that value.

Does the calculator find the rule from a list of terms?

No. Pattern inference can have many possible answers. Choose a defined sequence type and provide its first term and difference or ratio.