{"success":true,"version":"v1","request":{"tool":"get_calculator_schema","calculator_id":"geometric-sequence"},"result":{"entity_type":"calculator","id":"geometric-sequence","calculator_id":"geometric-sequence","canonical_url":"https://tttkmbb.com/math/geometric-sequence","name":"Geometric Sequence Calculator","title":"Geometric Sequence Calculator – nth Term, Sum of n Terms and Sum to Infinity","category":"math","category_name":"Math","tool_name":"calculate_geometric_sequence","featured_mcp_tool":false,"description":"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.","use_when":"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).","inputs":[{"name":"first_term","label":"First term (a₁)","type":"number","required":true,"description":"The first term of the sequence.","example":3},{"name":"common_ratio","label":"Common ratio (r)","type":"number","required":true,"description":"Factor between consecutive terms (may be negative or fractional).","example":2},{"name":"n","label":"Number of terms (n)","type":"integer","required":true,"min":1,"max":100000,"description":"Index of the term to compute and the number of terms to sum.","example":5}],"outputs":[{"name":"nth_term","label":"nth term (aₙ)","type":"number","decimals":8,"description":"a₁ × r^(n − 1)."},{"name":"sum","label":"Sum of first n terms (Sₙ)","type":"number","decimals":8,"description":"a₁ × (1 − rⁿ) / (1 − r), or n × a₁ when r = 1."},{"name":"infinite_sum","label":"Sum to infinity","type":"number","decimals":8,"description":"a₁ / (1 − r); only when |r| < 1 (the series converges)."},{"name":"converges","label":"Series converges","type":"boolean","decimals":4,"description":"true when |r| < 1, so the infinite sum exists."},{"name":"first_terms","label":"First terms","type":"number_list","decimals":8,"description":"The first min(n, 10) terms of the sequence."},{"name":"explicit_formula","label":"Explicit formula","type":"string","decimals":4,"description":"The formula for the general term with the given values."}],"input_schema":{"type":"object","properties":{"first_term":{"description":"The first term of the sequence.","type":"number","examples":[3]},"common_ratio":{"description":"Factor between consecutive terms (may be negative or fractional).","type":"number","examples":[2]},"n":{"description":"Index of the term to compute and the number of terms to sum.","type":"integer","minimum":1,"maximum":100000,"examples":[5]}},"additionalProperties":false,"required":["first_term","common_ratio","n"]},"output_schema":{"type":"object","properties":{"nth_term":{"description":"a₁ × r^(n − 1).","type":"number"},"sum":{"description":"a₁ × (1 − rⁿ) / (1 − r), or n × a₁ when r = 1.","type":"number"},"infinite_sum":{"description":"a₁ / (1 − r); only when |r| < 1 (the series converges).","type":"number"},"converges":{"description":"true when |r| < 1, so the infinite sum exists.","type":"boolean"},"first_terms":{"description":"The first min(n, 10) terms of the sequence.","type":"array","items":{"type":"number"}},"explicit_formula":{"description":"The formula for the general term with the given values.","type":"string"}}},"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","sources":[{"name":"Wikipedia – Geometric progression","url":"https://en.wikipedia.org/wiki/Geometric_progression","type":"reference","retrieved_at":"2026-09-23"},{"name":"Wikipedia – Geometric series","url":"https://en.wikipedia.org/wiki/Geometric_series","type":"reference","retrieved_at":"2026-09-23"},{"name":"Wolfram MathWorld – Geometric Series","url":"https://mathworld.wolfram.com/GeometricSeries.html","type":"reference","retrieved_at":"2026-09-23"}],"freshness":{"type":"static","max_age_seconds":null,"note":"Deterministic formula with fixed constants; results never go stale. Inputs supplied by the caller determine the output."},"examples":[{"name":"3, 6, 12, … (5 terms)","inputs":{"first_term":3,"common_ratio":2,"n":5},"expected":{"nth_term":48,"sum":93,"converges":false,"first_terms":[3,6,12,24,48],"explicit_formula":"a_n = 3 × 2^(n − 1)"},"url":"https://tttkmbb.com/api/v1/calculate/geometric-sequence?first_term=3&common_ratio=2&n=5"},{"name":"1, 1/2, 1/4, … (4 terms)","inputs":{"first_term":1,"common_ratio":0.5,"n":4},"expected":{"nth_term":0.125,"sum":1.875,"infinite_sum":2,"converges":true},"url":"https://tttkmbb.com/api/v1/calculate/geometric-sequence?first_term=1&common_ratio=0.5&n=4"}],"faq":[{"q":"When does the infinite sum exist?","a":"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?","a":"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)."}],"tags":["geometric sequence","geometric progression","geometric series","nth term","sum to infinity","common ratio"],"related":[{"calculator_id":"arithmetic-sequence","reason":"Sequences with a constant difference."},{"calculator_id":"compound-interest","reason":"Compound growth is a geometric sequence in the balance."},{"calculator_id":"exponent","reason":"The power rⁿ used in the formulas."}],"links":{"html":"https://tttkmbb.com/math/geometric-sequence","markdown":"https://tttkmbb.com/math/geometric-sequence.md","json":"https://tttkmbb.com/math/geometric-sequence.json","api":"https://tttkmbb.com/api/v1/calculate/geometric-sequence","schema":"https://tttkmbb.com/api/v1/calculators/geometric-sequence","openapi":"https://tttkmbb.com/openapi.json","mcp":"https://tttkmbb.com/mcp"},"version":"v1","updated_at":"2026-09-23"},"timestamp":"2026-09-23T23:24:45Z","next_actions":[{"tool":"run_calculator","calculator_id":"geometric-sequence","reason":"Run Geometric Sequence Calculator with the inputs above."}],"links":{"markdown":"https://tttkmbb.com/math/geometric-sequence.md"}}