Geometric Sequence Calculator

Geometric Sequence is evaluated from First Term, Common Ratio and Number of Terms. The calculation reports nth Term, Sum of n Terms and Sum to Infinity.

Results

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About the Geometric Sequence Calculator

Geometric Sequence is treated here as a quantitative relation between First Term, Common Ratio and Number of Terms and nth Term, Sum of n Terms, Sum to Infinity and Series Convergence.

The calculator uses a multi formula configuration. Each reported value is read as a direct evaluation of the stored rules with the declared field formats and units.

Formula basis:
aₙ = a₁ x r^(n - 1)
Sₙ = a₁ x (1 - rⁿ)/(1 - r)
S∞ = a₁/(1 - r) [only if |r| < 1]

Interpret the outputs in the order shown by the result fields. Optional inputs affect only the outputs that depend on those variables.

Formula & How It Works

The calculation applies the following relations exactly as recorded in the metadata:

aₙ = a₁ x r^(n - 1)
Sₙ = a₁ x (1 - rⁿ)/(1 - r)
S∞ = a₁/(1 - r) [only if |r| < 1]

Each output field is produced by substituting the supplied inputs into the relevant relation and then applying the declared rounding or text format.

Worked Examples

Example 1: Compound Interest — Balance Growth

Inputs

first_term: 1000 ratio: 1.08 n_terms: 10
nth Term: 1,999.0046271. Sum of n Terms: 14,486.56246591. Sum to Infinity: Divergent (|r| >= 1). Series Convergence: Divergent (|r| >= 1) - series grows without bound

With First Term = 1,000, Common Ratio = 1.08 and Number of Terms = 10 as the stated inputs, the result is nth Term = 1,999.0046271, Sum of n Terms = 14,486.56246591 and Sum to Infinity = Divergent (|r| >= 1). Each value corresponds to the declared output fields.

Example 2: Bouncing Ball — Convergent Series

Inputs

first_term: 10 ratio: 0.75 n_terms: 8
nth Term: 1.33483887. Sum of n Terms: 35.9954834. Sum to Infinity: 40. Series Convergence: Convergent (|r| = 0.75 < 1) - Sum Infinity = 40

With First Term = 10, Common Ratio = 0.75 and Number of Terms = 8 as the stated inputs, the result is nth Term = 1.33483887, Sum of n Terms = 35.9954834 and Sum to Infinity = 40. Each value corresponds to the declared output fields.

Example 3: Bacterial Growth — Doubling Time

Inputs

first_term: 100 ratio: 2 n_terms: 10
nth Term: 51,200. Sum of n Terms: 102,300. Sum to Infinity: Divergent (|r| >= 1). Series Convergence: Divergent (|r| >= 1) - series grows without bound

With First Term = 100, Common Ratio = 2 and Number of Terms = 10 as the stated inputs, the result is nth Term = 51,200, Sum of n Terms = 102,300 and Sum to Infinity = Divergent (|r| >= 1). Each value corresponds to the declared output fields.

Example 4: Zeno's Paradox — Infinite Series Sum

Inputs

first_term: 1 ratio: 0.5 n_terms: 10
nth Term: 0.00195313. Sum of n Terms: 1.99804688. Sum to Infinity: 2. Series Convergence: Convergent (|r| = 0.5 < 1) - Sum Infinity = 2

With First Term = 1, Common Ratio = 0.5 and Number of Terms = 10 as the stated inputs, the result is nth Term = 0.00195313, Sum of n Terms = 1.99804688 and Sum to Infinity = 2. Each value corresponds to the declared output fields.

Common Use Cases

  • Calculate compound interest growth as a geometric sequence
  • Find the nth term of 2, 6, 18, 54...
  • Compute sum of a geometric series
  • Determine if a series converges (|r| < 1)