04 // Interactive Formula Compendium // CALCULUS-ALGEBRAS_n = a₁ · (1 − rⁿ) / (1 − r), S_∞ = a₁ / (1 − r) (|r| < 1)
Geometric Sequence & Infinite Geometric Series Sum Formula
A geometric series sums terms where each entry is multiplied by a constant ratio r: for r ≠ 1, the finite sum of n terms is S_n = a₁(1 − rⁿ)/(1 − r), and when |r| < 1 the infinite series converges to S_∞ = a₁/(1 − r).
Formula Reference
Main Formula
S_n = a₁ · (1 − rⁿ) / (1 − r), S_∞ = a₁ / (1 − r) (|r| < 1)
Rearranged Forms
a_n = a₁ · rⁿ⁻¹ (n-th Term of Geometric Sequence)
r = a_{k+1} / a_k (Common Ratio)
G = √(a_{k−1} · a_{k+1}) (Geometric Mean of Neighbors)
Variables & Units
| Symbol | Name | Unit |
|---|---|---|
| S_n | Finite Geometric Series Sum (S_n) | dimensionless |
| a1 | First Term (a₁) | 1 |
| r | Common Ratio (r) | 1 |
| n | Number of Terms (n) | integer |
Inputs & Parameters
[1]
[1]
[integer]
Calculated Result (Finite Geometric Series Sum (S_n))
S_n =199.8047dimensionless
Formula Notes & Background
- Write S_n = a₁ + a₁r + a₁r² + … + a₁rⁿ⁻¹ and multiply both sides by r: rS_n = a₁r + a₁r² + … + a₁rⁿ.
- Subtract the second equation from the first so all interior terms telescope: S_n(1 − r) = a₁ − a₁rⁿ.
- Divide by (1 − r) for r ≠ 1; if |r| < 1, taking lim_{n→∞} rⁿ = 0 gives S_∞ = a₁ / (1 − r).
Step-by-Step Examples
#1: Find the sum of the first 10 terms of 100 + 50 + 25 + 12.5 + …
a₁ = 100r = 0.5n = 10
S₁₀ = 100 × (1 − 0.5¹⁰) / (1 − 0.5) = 200 × (1 − 1/1024)
→ S₁₀ = 199.8047 (and S_∞ = 100 / 0.5 = 200)
#2: Convert the repeating decimal 0.363636… into an exact fraction.
a₁ = 0.36r = 0.01n → ∞
S_∞ = 0.36 / (1 − 0.01) = 0.36 / 0.99 = 36 / 99
→ 4 / 11
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Frequently Asked Questions
Why is the geometric series formula the foundation of all loan and annuity math?+
The present value of n equal cash flows discounted at rate i per period is a geometric series with ratio r = 1/(1 + i).
What happens if |r| ≥ 1 in an infinite geometric series?+
Since rⁿ does not approach 0 as n → ∞, the general term fails the divergence test and the infinite sum diverges.
What does S_n equal when the common ratio is exactly r = 1?+
Every term equals a₁, so the sum of n identical terms is simply S_n = n · a₁.