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Geometric Sequence Calculator

Q: When does the infinite sum exist?

Only when |r| < 1; then the terms shrink toward zero and the partial sums approach a₁ / (1 − r). For |r| ≥ 1 the series diverges and infinite_sum is omitted.

Q: What if the ratio is negative?

Terms alternate in sign. The formulas still apply, and the series converges when |r| < 1 (e.g. r = −0.5 gives S∞ = a₁ / 1.5).

Computes the n-th term and the sum of the first n terms of a geometric progression from the first term and common ratio, plus the sum to infinity when |r| < 1.

When to use

Each term is the previous one multiplied by a constant ratio (e.g. 3, 6, 12, …) and you need a term, a partial sum or the limit of the series.

Do not use when: Terms change by a constant difference (use arithmetic-sequence) or you need a money value with periodic deposits (use future-value / compound-interest).

Formula

aₙ = a₁·r^(n−1); Sₙ = a₁·(1 − rⁿ)/(1 − r) for r ≠ 1 (Sₙ = n·a₁ for r = 1); S∞ = a₁/(1 − r) for |r| < 1

Inputs

Parameter	Type	Required	Description
`first_term`	number	yes	The first term of the sequence.
`common_ratio`	number	yes	Factor between consecutive terms (may be negative or fractional).
`n`	integer	yes	Index of the term to compute and the number of terms to sum. Range: ≥ 1, ≤ 100000

Outputs

Output	Type	Description
`nth_term`	number	a₁ × r^(n − 1).
`sum`	number	a₁ × (1 − rⁿ) / (1 − r), or n × a₁ when r = 1.
`infinite_sum`	number	a₁ / (1 − r); only when \|r\| < 1 (the series converges).
`converges`	boolean	true when \|r\| < 1, so the infinite sum exists.
`first_terms`	number_list	The first min(n, 10) terms of the sequence.
`explicit_formula`	string	The formula for the general term with the given values.

Example

3, 6, 12, … (5 terms): {"first_term":3,"common_ratio":2,"n":5} → {"nth_term":48,"sum":93,"converges":false,"first_terms":[3,6,12,24,48],"explicit_formula":"a_n = 3 × 2^(n − 1)"}

1, 1/2, 1/4, … (4 terms): {"first_term":1,"common_ratio":0.5,"n":4} → {"nth_term":0.125,"sum":1.875,"infinite_sum":2,"converges":true}

GET https://tttkmbb.com/api/v1/calculate/geometric-sequence?first_term=3&common_ratio=2&n=5

Machine access

API: GET https://tttkmbb.com/api/v1/calculate/geometric-sequence (query parameters) or POST with a JSON body {"inputs": {...}}
Schema: https://tttkmbb.com/api/v1/calculators/geometric-sequence · Markdown: https://tttkmbb.com/math/geometric-sequence.md · JSON definition: https://tttkmbb.com/math/geometric-sequence.json
MCP: server https://tttkmbb.com/mcp, tool run_calculator with calculator_id="geometric-sequence"
OpenAPI operationId: calculate_geometric_sequence
Freshness: static. Authentication: none. Rate limit: fair use (see rate limits).

Sources

Wikipedia – Geometric progression (reference)
Wikipedia – Geometric series (reference)
Wolfram MathWorld – Geometric Series (reference)

FAQ

When does the infinite sum exist?

Only when |r| < 1; then the terms shrink toward zero and the partial sums approach a₁ / (1 − r). For |r| ≥ 1 the series diverges and infinite_sum is omitted.

What if the ratio is negative?

Terms alternate in sign. The formulas still apply, and the series converges when |r| < 1 (e.g. r = −0.5 gives S∞ = a₁ / 1.5).

Related calculators

Arithmetic Sequence Calculator — Sequences with a constant difference.
Compound Interest Calculator — Compound growth is a geometric sequence in the balance.
Exponent Calculator — The power rⁿ used in the formulas.