Geometric Sequence Calculator Formula

Understand the math behind the geometric sequence calculator. Each variable explained with a worked example.

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Formulas Used

Nth Term

nth_term = a1 * pow(r, n - 1)

Sum Finite

sum_finite = r != 1 ? a1 * (1 - pow(r, n)) / (1 - r) : a1 * n

Sum Infinite

sum_infinite = abs(r) < 1 ? a1 / (1 - r) : 0

Variables

VariableDescriptionDefault
a1First Term (a₁)2
rCommon Ratio (r)3
nNumber of Terms (n)6

How It Works

Geometric Sequence

nth Term

aₙ = a₁ × r^(n-1)

Sum of n Terms

Sₙ = a₁ × (1 - r^n) / (1 - r) (when r ≠ 1)

Infinite Sum (converges only when |r| < 1)

S∞ = a₁ / (1 - r)

A geometric sequence has a constant ratio between consecutive terms.

Worked Example

Geometric sequence: first term = 2, common ratio = 3, 6 terms.

a1 = 2r = 3n = 6
  1. 01a₆ = 2 × 3⁵ = 2 × 243 = 486
  2. 02Sum = 2 × (1 - 3⁶) / (1 - 3) = 2 × (1 - 729) / (-2) = 728
  3. 03Sequence: 2, 6, 18, 54, 162, 486

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