{"success":true,"version":"v1","request":{"tool":"get_calculator_schema","calculator_id":"arithmetic-sequence"},"result":{"entity_type":"calculator","id":"arithmetic-sequence","calculator_id":"arithmetic-sequence","canonical_url":"https://tttkmbb.com/math/arithmetic-sequence","name":"Arithmetic Sequence Calculator","title":"Arithmetic Sequence Calculator – nth Term and Sum of an Arithmetic Progression","category":"math","category_name":"Math","tool_name":"calculate_arithmetic_sequence","featured_mcp_tool":false,"description":"Computes the n-th term and the sum of the first n terms of an arithmetic progression from the first term and the common difference, and lists the first terms.","use_when":"Terms increase or decrease by a constant amount (e.g. 2, 5, 8, …) and you need a specific term or the total of the first n terms.","do_not_use_when":"Terms are multiplied by a constant ratio (use geometric-sequence) or you need interest that compounds (use compound-interest).","inputs":[{"name":"first_term","label":"First term (a₁)","type":"number","required":true,"description":"The first term of the sequence.","example":2},{"name":"common_difference","label":"Common difference (d)","type":"number","required":true,"description":"Amount added to each term to get the next (negative for a decreasing sequence).","example":3},{"name":"n","label":"Number of terms (n)","type":"integer","required":true,"min":1,"max":1000000000,"description":"Index of the term to compute and the number of terms to sum.","example":10}],"outputs":[{"name":"nth_term","label":"nth term (aₙ)","type":"number","decimals":6,"description":"a₁ + (n − 1) × d."},{"name":"sum","label":"Sum of first n terms (Sₙ)","type":"number","decimals":6,"description":"n × (a₁ + aₙ) / 2."},{"name":"first_terms","label":"First terms","type":"number_list","decimals":6,"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":[2]},"common_difference":{"description":"Amount added to each term to get the next (negative for a decreasing sequence).","type":"number","examples":[3]},"n":{"description":"Index of the term to compute and the number of terms to sum.","type":"integer","minimum":1,"maximum":1000000000,"examples":[10]}},"additionalProperties":false,"required":["first_term","common_difference","n"]},"output_schema":{"type":"object","properties":{"nth_term":{"description":"a₁ + (n − 1) × d.","type":"number"},"sum":{"description":"n × (a₁ + aₙ) / 2.","type":"number"},"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 + (n − 1)·d; Sₙ = n·(a₁ + aₙ)/2 = n·(2a₁ + (n − 1)·d)/2","sources":[{"name":"Wikipedia – Arithmetic progression","url":"https://en.wikipedia.org/wiki/Arithmetic_progression","type":"reference","retrieved_at":"2026-09-23"},{"name":"Wolfram MathWorld – Arithmetic Series","url":"https://mathworld.wolfram.com/ArithmeticSeries.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":"2, 5, 8, … (10 terms)","inputs":{"first_term":2,"common_difference":3,"n":10},"expected":{"nth_term":29,"sum":155,"first_terms":[2,5,8,11,14,17,20,23,26,29],"explicit_formula":"a_n = 2 + (n − 1) × 3"},"url":"https://tttkmbb.com/api/v1/calculate/arithmetic-sequence?first_term=2&common_difference=3&n=10"},{"name":"5, 3, 1, … (6 terms)","inputs":{"first_term":5,"common_difference":-2,"n":6},"expected":{"nth_term":-5,"sum":0,"first_terms":[5,3,1,-1,-3,-5]},"url":"https://tttkmbb.com/api/v1/calculate/arithmetic-sequence?first_term=5&common_difference=-2&n=6"}],"faq":[{"q":"How do I find d from two terms?","a":"d = (a_m − a_k) / (m − k). For a₁ = 2 and a₄ = 11, d = (11 − 2) / 3 = 3."},{"q":"Can the sum be zero or negative?","a":"Yes. With a negative common difference the terms eventually become negative and the partial sums can decrease, as in 5, 3, 1, −1, −3, −5 whose sum is 0."}],"tags":["arithmetic sequence","arithmetic progression","nth term","sum of terms","common difference"],"related":[{"calculator_id":"geometric-sequence","reason":"Sequences with a constant ratio instead of a constant difference."},{"calculator_id":"simple-interest","reason":"Simple interest grows linearly like an arithmetic sequence."}],"links":{"html":"https://tttkmbb.com/math/arithmetic-sequence","markdown":"https://tttkmbb.com/math/arithmetic-sequence.md","json":"https://tttkmbb.com/math/arithmetic-sequence.json","api":"https://tttkmbb.com/api/v1/calculate/arithmetic-sequence","schema":"https://tttkmbb.com/api/v1/calculators/arithmetic-sequence","openapi":"https://tttkmbb.com/openapi.json","mcp":"https://tttkmbb.com/mcp"},"version":"v1","updated_at":"2026-09-23"},"timestamp":"2026-09-23T23:22:57Z"}