Geometric Series Sum Calculator

Definition

The finite sum of a geometric sequence is Sₙ = a₁ + a₂ + … + aₙ, where each next term multiplies the previous one by a constant common ratio q. With a₁ given and aₖ₊₁ = q · aₖ, the terms are aₖ = a₁ · qᵏ⁻¹. Their sum has the closed form Sₙ = a₁(1 − qⁿ)/(1 − q) for q ≠ 1. The algebraically identical form Sₙ = a₁(qⁿ − 1)/(q − 1) differs only by a sign in the numerator and denominator.

This calculator proceeds exactly as follows: when |q − 1| < 10⁻¹² it sets q = 1 and computes Sₙ = a₁ · n; otherwise Sₙ = a₁(1 − q^(n))/(1 − q). The term count n comes from parseTermCount: a finite whole number ≥ 1. That is not n = max(1, floor(n)) from the nth-term page.

An infinite geometric series is a separate topic: for |q| < 1 the limit is S = a₁/(1 − q). Here n is finite and always an integer. The calculator does not solve for q from a sum, solve for n, or return a single term aₙ.

Historical and conceptual context

Sums of geometric progressions appear in ancient computational texts — doubling, halving, proportional “percent” ideas. Zeno’s paradox (Achilles and the tortoise) connects to an infinite geometric series; secondary school usually starts with the finite sum Sₙ before the limit. Euclid’s Elements develops proportion theory; modern algebra writes the same idea as a closed formula with the power qⁿ.

Applications include the present value of an annuity with a fixed discount factor, compound-interest payment streams, population growth with a constant factor, signal filtering, and power series. In discrete finance the factor (1+i) plays the role of q. The name “geometric” ties to the geometric mean of positive terms, but this page’s result is a sum, not a mean.

Distinguish three neighbors: the nth term aₙ (one term), the arithmetic series sum (constant difference r), and the geometric series sum (constant ratio q). This page belongs to the third world and does not replace either of the other models.

In the school curriculum, the sum Sₙ appears after the definition of a geometric sequence and before the infinite-series limit. Students usually list a few terms first, then learn the closed formula, and only later meet the |q| < 1 condition for convergence. This calculator covers the middle stage: finite n and a ready Sₙ.

Formal properties

Domain: finite real a₁ and q; integer n ≥ 1. The result Sₙ must be a finite floating-point number. With q = 1 every term equals a₁ and Sₙ = n·a₁. With q = −1 terms alternate ±a₁ and the sum depends on the parity of n. When |q| > 1 the magnitude of the sum grows quickly; when 0 < |q| < 1 later terms shrink and the sum approaches a₁/(1 − q), but the calculator still evaluates finite n, not the limit.

Identity: a₁(1 − qⁿ)/(1 − q) = a₁(qⁿ − 1)/(q − 1) for q ≠ 1. Scaling: the sum of (c·aₖ) is c·Sₙ. With a₁ = 0 the sum is zero for finite inputs.

The last term aₙ = a₁·qⁿ⁻¹ appears in the panel as a check; it does not replace Sₙ. Bars illustrate |aₖ| for the first min(n, 8) terms.

Full derivation

Step 1. Write S = a₁ + a₁q + a₁q² + … + a₁qⁿ⁻¹.

Step 2. Multiply by q: qS = a₁q + a₁q² + … + a₁qⁿ.

Step 3. Subtract: S − qS = a₁ − a₁qⁿ, so S(1 − q) = a₁(1 − qⁿ).

Step 4. For q ≠ 1: S = a₁(1 − qⁿ)/(1 − q). For q = 1 return to S = n·a₁.

Step 5. Example 2, 3, 4: qⁿ = 81, S = 2(1 − 81)/(1 − 3) = 2(−80)/(−2) = 80. Example 5, 1, 6: S = 5·6 = 30. Example 4, −2, 5: qⁿ = −32, S = 4(1 − (−32))/(1 − (−2)) = 4·33/3 = 44.

Special cases

2, 3, 4 → 80. 5, 1, 6 → 30. 4, −2, 5 → 44. 1, 2, 8 → 255. 3, 3, 3 → 39. 16, 0.5, 4 → 30. 7, 2, 1 → 7 (single term). Entering n = 4.5 fails validation — the term count must be a whole number.

Very large |q| and n may yield Infinity in floating-point arithmetic — the calculator reports an out-of-range error.

Numerical notes

Computation: q^(n) and division by (1 − q) in the browser’s floating-point arithmetic, with a separate branch when |q − 1| < 10⁻¹². For ordinary school examples the displayed value matches the decimal intent; extreme powers meet floating-point rounding. This is an educational tool, not a CAS.

The calculator accepts a comma or a period as the decimal separator. Formatting follows the page language, without a forced six-decimal trim.

Bars show the first min(n, 8) terms; when n > 8 the full Sₙ still appears in the panel.

Related notions and sources

Related: geometric sequence term, arithmetic series sum, compound interest, geometric mean, infinite series for |q| < 1. Sources: Wikipedia — Geometric series; MathWorld — Geometric Series; school algebra Sₙ = a₁(1−qⁿ)/(1−q). Educational tool.