# Question 31881: Successor Recurrence for Factorial [Return to catalogue page 160](https://ocme.evemisslab.com/en/questions/page/160/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20260926-factorial-foundations/#q-20260926-factorial-successor-recurrence-0006) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0062-31881 · q-20260926-factorial-successor-recurrence-0006 ## Original question and givens Given 5!=120, use (n+1)!=(n+1)×n! to find 6!. - G001 — Given 5!=120, use (n+1)!=(n+1)×n! to find 6!. ## Definition and scope Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate. - [mko-factorial](https://ocme.evemisslab.com/en/mko/mko-factorial/) ## Step-by-step derivation S001 · impliesBefore source roles = [qf-factorial-successor-recurrence, no-domain-field, successor_recurrence, no-form-field] After derived = [[exact_integer_STRING]] Conditions: none Check THIS bank interface:BASE equations use form/domain;BASE events/complement have no operation;factorials have no domain.ext18 uses local hyphens,other legacy operations underscores.Do not force every answer into a tuple or fraction. Step identifiers and dependenciesRule: R62_EXACT_source_keys_domain_operation_form_AND_output_types · Uses: G001 S002 · calculationBefore Factorial requested n ORIGINAL factorial factor2 := mul(1, 2) After Factorial requested n ORIGINAL factorial factor2 = 2 Conditions: none Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore Factorial requested n ORIGINAL factorial factor3 := mul(2, 3) After Factorial requested n ORIGINAL factorial factor3 = 6 Conditions: none Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore Factorial requested n ORIGINAL factorial factor4 := mul(6, 4) After Factorial requested n ORIGINAL factorial factor4 = 24 Conditions: none Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore Factorial requested n ORIGINAL factorial factor5 := mul(24, 5) After Factorial requested n ORIGINAL factorial factor5 = 120 Conditions: none Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · impliesBefore source roles = [Factorial requested n, 5] After derived = [120] Conditions: none Factorial multiplies the COMPLETE2..n factors;at n0/1 the empty product is1.Use arbitrary-precision integers,not Number/scientific notation or another family n64 cap. Step identifiers and dependenciesRule: Factorial_COMPLETE_product_EMPTY0and1_are1_not_input_n64_cap · Uses: G001, S005 S007 · impliesBefore source roles = [5, 120] After derived = [120] Conditions: none Recompute the WHOLE n! and compare the GIVEN decimal STRING first;do not treat an unchecked hint as an answer oracle.Require the exact string,retaining every digit. Step identifiers and dependenciesRule: Factorial_GIVEN_nfactorial_STRING_verified_not_assumed · Uses: G001, S006 S008 · calculationBefore Factorial ORIGINAL successor index n plus1 := sum(5, 1) After Factorial ORIGINAL successor index n plus1 = 6 Conditions: none Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore Factorial ORIGINAL successor factor times verified n factorial := mul(6, 120) After Factorial ORIGINAL successor factor times verified n factorial = 720 Conditions: none Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · impliesBefore source roles = [5, 120, 6] After derived = [720] Conditions: none AFTER verifying n!,apply(n+1)!=(n+1)n! by one exact multiplication.At n0 this gives1!=1,not0;do not exchange n!and(n+1)!. Step identifiers and dependenciesRule: Factorial_SUCCESSOR_recurrence_uses_verified_nfactorial · Uses: G001, S009 ## Reverse checks - S010 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate. - S009 · premise_and_result_audit — Recompute Factorial ORIGINAL successor factor times verified n factorial with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute Factorial ORIGINAL successor index n plus1 with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate. - S006 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate. - S005 · premise_and_result_audit — Recompute Factorial requested n ORIGINAL factorial factor5 with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute Factorial requested n ORIGINAL factorial factor4 with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute Factorial requested n ORIGINAL factorial factor3 with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute Factorial requested n ORIGINAL factorial factor2 with exact rational arithmetic, not rounded decimals. - S001 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate. ## Result 720 The full substitutions and premise checks give the original answer; source identity,format and status are unchanged. ## This question review and source binding Review date: 2026-10-26 Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 3f3d8c4fda10c27efa97d1736a630d593948476936eb0e7cf3d460c298d56caa - candidate record SHA-256: f400da55b4b3f88ed6bae57f7d5dfd73c75dec4b47f38646d29c70aab00b4ec3 - semantic review receipt SHA-256: 963099ffc9f754069fdce6792949a85ed350472465eaceb3038fa8011cf672bd - question-first witness SHA-256: d7655324c5d11528e1af90b757384cebd3313e19dc5f6f5b9b8ea7624fb06f6f - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20260926-factorial-successor-recurrence-0006.json)